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Consider the following expressions:

6 - 1/4 * 28 - 2 1/4 __ 3/4 + 4 * 2 - 11.
Which operator can be inserted to make the statement true? (1 point)
>
≥
<
=

55 answers

To determine which operator can be inserted to make the statement true, we first need to simplify both sides of the expression.

Let's break it down step by step:

Left Side:

  1. Start with the expression: \[ 6 - \frac{1}{4} \times 28 - 2 \frac{1}{4} \]

  2. Convert \( 2 \frac{1}{4} \) to an improper fraction: \[ 2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

  3. Now the left side becomes: \[ 6 - \frac{1}{4} \times 28 - \frac{9}{4} \]

  4. Calculate \( \frac{1}{4} \times 28 \): \[ \frac{1}{4} \times 28 = 7 \]

  5. Substitute that back: \[ 6 - 7 - \frac{9}{4} \]

  6. Combine \( 6 - 7 = -1 \): \[ -1 - \frac{9}{4} \]

  7. To combine these, convert \(-1\) to a fraction: \[ -1 = -\frac{4}{4} \]

  8. Now combine: \[ -\frac{4}{4} - \frac{9}{4} = -\frac{13}{4} \]

So, the left side equals: \[ -\frac{13}{4} \]

Right Side:

  1. Now simplify the right side: \[ 3/4 + 4 \times 2 - 11 \]

  2. Calculate \( 4 \times 2 = 8 \), so the right side becomes: \[ 3/4 + 8 - 11 \]

  3. Combine \( 8 - 11 = -3 \): \[ 3/4 - 3 \]

  4. Convert \(-3\) to a fraction: \[ -3 = -\frac{12}{4} \]

  5. Combine: \[ 3/4 - \frac{12}{4} = \frac{3 - 12}{4} = \frac{-9}{4} \]

So, the right side equals: \[ -\frac{9}{4} \]

Final Comparison: Now we compare the two sides:

  • Left side: \( -\frac{13}{4} \)
  • Right side: \( -\frac{9}{4} \)

Now let's see which operator makes it true:

  • \( -\frac{13}{4} < -\frac{9}{4} \)

Thus, the operator that can be inserted to make the statement true is: "<"

Consider the following expressions:

6 - 1/4 * 28 - 2 1/4 __ 3/4 + 4 * 2 - 11.
Which operator can be inserted to make the statement true? (1 point)
>
≥
<
=

To determine which operator can be inserted to make the statement true, we need to evaluate both sides of the expression carefully.

Let's rewrite and simplify each side of the expression step-by-step.

Left Side: \( 6 - \frac{1}{4} * 28 - 2\frac{1}{4} \)

  1. Let's break down \( 2\frac{1}{4} \): \[ 2\frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

  2. The left-hand expression then becomes: \[ 6 - \frac{1}{4} \times 28 - \frac{9}{4} \]

  3. Calculate \(\frac{1}{4} \times 28\): \[ \frac{1}{4} \times 28 = 7 \]

  4. Substitute that back: \[ 6 - 7 - \frac{9}{4} \]

  5. Calculate \(6 - 7\): \[ 6 - 7 = -1 \]

  6. Combine \(-1 - \frac{9}{4}\): Convert \(-1\) to a fraction: \[ -1 = -\frac{4}{4} \] So, \[ -\frac{4}{4} - \frac{9}{4} = -\frac{13}{4} \]

Now we have: \[ \text{Left side} = -\frac{13}{4} \]

Right Side: \( \frac{3}{4} + 4 \times 2 - 11 \)

  1. Start by simplifying \( 4 \times 2 \): \[ 4 \times 2 = 8 \]

  2. So the right-hand expression becomes: \[ \frac{3}{4} + 8 - 11 \]

  3. Combine \( 8 - 11 \): \[ 8 - 11 = -3 \]

  4. Now the right side simplifies to: \[ \frac{3}{4} - 3 \] Convert \(-3\) to a fraction: \[ -3 = -\frac{12}{4} \] So, \[ \frac{3}{4} - \frac{12}{4} = \frac{3 - 12}{4} = \frac{-9}{4} \]

Now we have: \[ \text{Right side} = -\frac{9}{4} \]

Comparing Both Sides:

  • Left side: \( -\frac{13}{4} \)
  • Right side: \( -\frac{9}{4} \)

Now we can compare these values:

  • \( -\frac{13}{4} < -\frac{9}{4} \)

Conclusion

The appropriate operator to insert is: \[ < \]

So the answer is \( < \).

Which of the following values for b makes the inequality -32.5 - 6b ≥ 48.5 true?
-13.5
-12
-13
-12.5

To determine which value for \( b \) makes the inequality \( -32.5 - 6b \geq 48.5 \) true, we can solve the inequality step-by-step.

Step 1: Isolate \( b \)

  1. Start with the original inequality: \[ -32.5 - 6b \geq 48.5 \]

  2. Add \( 32.5 \) to both sides: \[ -6b \geq 48.5 + 32.5 \]

  3. Calculate the right-hand side: \[ -6b \geq 81 \]

  4. Now, divide both sides by \(-6\). Remember, dividing by a negative number flips the inequality sign: \[ b \leq -\frac{81}{6} \]

  5. Simplifying \(-\frac{81}{6}\): \[ -\frac{81}{6} = -13.5 \] So, \[ b \leq -13.5 \]

Step 2: Check the given values

We need to see which of the provided values satisfy \( b \leq -13.5 \):

  • -13.5: \( -13.5 \leq -13.5 \) (True)
  • -12: \( -12 > -13.5 \) (False)
  • -13: \( -13 > -13.5 \) (False)
  • -12.5: \( -12.5 > -13.5 \) (False)

Conclusion

The only value that makes the inequality true is: \[ \boxed{-13.5} \]

Which of the following values for b makes the inequality -32.5 - 6b ≥ 48.5 true?

To determine which value for \( b \) makes the inequality

\[ -32.5 - 6b \geq 48.5 \]

true, we'll solve the inequality step-by-step.

Step 1: Isolate \( b \)

  1. Start with the original inequality: \[ -32.5 - 6b \geq 48.5 \]

  2. Add \( 32.5 \) to both sides: \[ -6b \geq 48.5 + 32.5 \]

  3. Calculate the right-hand side: \[ -6b \geq 81 \]

  4. Now, divide both sides by \(-6\). Remember, when dividing by a negative number, the inequality sign flips: \[ b \leq -\frac{81}{6} \]

  5. Simplifying \(-\frac{81}{6}\): \[ -\frac{81}{6} = -13.5 \]

Conclusion

So we have the inequality: \[ b \leq -13.5 \]

Now you can pick values for \( b \) that satisfy this inequality. If the question included specific values for \( b \) (like -14, -13, -12, etc.), only the values less than or equal to -13.5 would make the inequality true.

If you provide the specific values for \( b \), I can confirm which of those makes the inequality true!

Which of the following equations has the solution set {-3} ?

To find an equation whose solution set is \({-3}\), we want to construct an equation that is satisfied only when \( x = -3 \).

One straightforward way to create such an equation is to construct a linear equation. For instance, you could take the following approaches:

  1. Basic Equation Format: You could create an equation in the form of: \[ x + 3 = 0 \] Here, if we substitute \( x = -3 \): \[ -3 + 3 = 0 \quad \text{(True)} \] Thus, the solution set is \({-3}\).

  2. Quadratic Equation: Another example could be a quadratic equation: \[ (x + 3)^2 = 0 \] This equation is satisfied when \( x + 3 = 0 \) (which also gives \( x = -3 \)), resulting in the solution set again being \({-3}\).

  3. Inequality Format: You can also create an inequality: \[ |x + 3| = 0 \] This is satisfied only when \( x + 3 = 0 \) or \( x = -3 \).

If you have a specific list of equations you'd like me to evaluate, please provide that list, and I can confirm which one has the solution set \({-3}\).

specific list of equations
4x - 8 = 4
4x - 8 = -4
4x + 8 = 4
4x + 8 = -4

To determine which of the given equations has the solution set \({-3}\), we can substitute \(x = -3\) into each equation and check which one holds true.

Given Equations:

  1. Equation 1: \[ 4x - 8 = 4 \] Substituting \(x = -3\): \[ 4(-3) - 8 = 4 \quad \Rightarrow \quad -12 - 8 = 4 \quad \Rightarrow \quad -20 \neq 4 \quad (\text{False}) \]

  2. Equation 2: \[ 4x - 8 = -4 \] Substituting \(x = -3\): \[ 4(-3) - 8 = -4 \quad \Rightarrow \quad -12 - 8 = -4 \quad \Rightarrow \quad -20 \neq -4 \quad (\text{False}) \]

  3. Equation 3: \[ 4x + 8 = 4 \] Substituting \(x = -3\): \[ 4(-3) + 8 = 4 \quad \Rightarrow \quad -12 + 8 = 4 \quad \Rightarrow \quad -4 \neq 4 \quad (\text{False}) \]

  4. Equation 4: \[ 4x + 8 = -4 \] Substituting \(x = -3\): \[ 4(-3) + 8 = -4 \quad \Rightarrow \quad -12 + 8 = -4 \quad \Rightarrow \quad -4 = -4 \quad (\text{True}) \]

Conclusion

The equation that has the solution set \({-3}\) is: \[ \boxed{4x + 8 = -4} \]

Which set expresses the solution to the compound inequality -2 ≤ 2x + 1/3 - 4 < 1?

To solve the compound inequality \(-2 \leq 2x + \frac{1}{3} - 4 < 1\), we'll break it down into two parts and simplify step by step.

Step 1: Solve the Left Part

Starting with: \[ -2 \leq 2x + \frac{1}{3} - 4 \]

First, we can simplify the right side: \[ -2 \leq 2x - 4 + \frac{1}{3} \]

Combine \(-4\) and \(\frac{1}{3}\) by converting \(-4\) to a fraction: \[ -4 = -\frac{12}{3} \] So we have: \[ -2 \leq 2x - \frac{12}{3} + \frac{1}{3} \]

This simplifies to: \[ -2 \leq 2x - \frac{11}{3} \]

Now, isolate \(2x\):

  1. Add \(\frac{11}{3}\) to both sides: \[ -2 + \frac{11}{3} \leq 2x \] Convert \(-2\) to a fraction: \[ -2 = -\frac{6}{3} \] Thus: \[ -\frac{6}{3} + \frac{11}{3} \leq 2x \] Which results in: \[ \frac{5}{3} \leq 2x \]

  2. Divide both sides by \(2\): \[ \frac{5}{6} \leq x \quad \text{(or, equivalently, } x \geq \frac{5}{6} \text{)} \]

Step 2: Solve the Right Part

Now let's solve: \[ 2x + \frac{1}{3} - 4 < 1 \]

Again, simplify the right side:

  1. Combine: \[ 2x - 4 + \frac{1}{3} < 1 \] \[ 2x - \frac{12}{3} + \frac{1}{3} < 1 \] Which simplifies to: \[ 2x - \frac{11}{3} < 1 \]

  2. Add \(\frac{11}{3}\) to both sides: \[ 2x < 1 + \frac{11}{3} \] Convert \(1\) to a fraction: \[ 1 = \frac{3}{3} \] Thus: \[ 2x < \frac{3}{3} + \frac{11}{3} \] This simplifies to: \[ 2x < \frac{14}{3} \]

  3. Divide both sides by \(2\): \[ x < \frac{14}{6} = \frac{7}{3} \]

Step 3: Combine the Results

From both parts, we have: \[ \frac{5}{6} \leq x < \frac{7}{3} \]

In interval notation, the solution can be expressed as: \[ \left[\frac{5}{6}, \frac{7}{3}\right) \]

Conclusion

The solution set to the compound inequality \(-2 \leq 2x + \frac{1}{3} - 4 < 1\) is: \[ \boxed{\left[\frac{5}{6}, \frac{7}{3}\right)} \]

Which set expresses the solution to the compound inequality -2 ≤ (2x + 1)/3 - 4 < 1?

To solve the compound inequality

\[ -2 \leq \frac{2x + 1}{3} - 4 < 1, \]

we'll break it into two parts and simplify step by step.

Step 1: Solve the Left Part

Starting with the left part:

\[ -2 \leq \frac{2x + 1}{3} - 4. \]

First, isolate the fraction by adding \(4\) to both sides:

\[ -2 + 4 \leq \frac{2x + 1}{3}. \]

This simplifies to:

\[ 2 \leq \frac{2x + 1}{3}. \]

Next, eliminate the fraction by multiplying both sides by \(3\):

\[ 3 \times 2 \leq 2x + 1, \]

which simplifies to:

\[ 6 \leq 2x + 1. \]

Now, subtract \(1\) from both sides:

\[ 6 - 1 \leq 2x, \]

or:

\[ 5 \leq 2x. \]

Dividing both sides by \(2\):

\[ \frac{5}{2} \leq x \quad \text{(or equivalently } x \geq \frac{5}{2}\text{)}. \]

Step 2: Solve the Right Part

Now, for the right part:

\[ \frac{2x + 1}{3} - 4 < 1. \]

Again, isolate the fraction by adding \(4\) to both sides:

\[ \frac{2x + 1}{3} < 1 + 4. \]

This simplifies to:

\[ \frac{2x + 1}{3} < 5. \]

Now, eliminate the fraction by multiplying both sides by \(3\):

\[ 2x + 1 < 15. \]

Subtracting \(1\) from both sides gives:

\[ 2x < 15 - 1, \]

which simplifies to:

\[ 2x < 14. \]

Dividing both sides by \(2\):

\[ x < 7. \]

Step 3: Combine the Results

From the two parts, we found:

  1. \(x \geq \frac{5}{2}\)
  2. \(x < 7\)

Combining these results, we can express the solution in interval notation:

\[ \left[\frac{5}{2}, 7\right). \]

Conclusion

The set that expresses the solution to the compound inequality

\[ -2 \leq \frac{2x + 1}{3} - 4 < 1 \]

is

\[ \boxed{\left[\frac{5}{2}, 7\right)}. \]

What property of equality is displayed in step 3 in the solving of 9m + 4 - 7m = (5m+7)/3 ?

Step 1:
9m + 4 - 7m = (5m + 7)/3
9m - 7m + 4 = (5m + 7)/3

Step 2:
2m + 4 = (5m + 7)/3
3(2m + 4) = ((5m + 7/3) 3
Step 3:
6m + 12 = 5m + 7
6m + 12 - 12 = 5m + 7 - 12
Step 4:
6m = 5m - 5
6m - 5m = 5m - 5m - 5
m = -5

In the given steps of solving the equation \(9m + 4 - 7m = \frac{5m + 7}{3}\), Step 3 involves manipulating both sides of the equation:

Step 3 Recap:

\[ 6m + 12 = 5m + 7 \] From there, you subtract \(12\) from both sides: \[ 6m + 12 - 12 = 5m + 7 - 12 \]

This yields: \[ 6m = 5m - 5 \]

Property of Equality:

The property displayed in Step 3 is the Subtraction Property of Equality.

The Subtraction Property of Equality states that if you have an equation and you subtract the same value from both sides of the equation, the equality remains balanced.

Conclusion:

In this case, you subtracted \(12\) from both sides, thereby preserving the equality, which is a clear demonstration of the Subtraction Property of Equality.

Which linear equation has the same solution set as (9d)/4 + 3 - 2d = -2 * (3 + d)?

To find a linear equation that has the same solution set as the equation

\[ \frac{9d}{4} + 3 - 2d = -2(3 + d), \]

we will simplify the given equation.

Step 1: Simplify the Right Side

First, simplify the right side:

\[ -2(3 + d) = -6 - 2d. \]

Step 2: Rewrite the Equation

Now we can rewrite the equation:

\[ \frac{9d}{4} + 3 - 2d = -6 - 2d. \]

Step 3: Add \(2d\) to Both Sides

To eliminate \(-2d\) from both sides, we can add \(2d\) to both sides:

\[ \frac{9d}{4} + 3 = -6. \]

Step 4: Isolate the Fractions

Next, subtract \(3\) from both sides:

\[ \frac{9d}{4} = -6 - 3, \]

which simplifies to:

\[ \frac{9d}{4} = -9. \]

Step 5: Eliminate the Fraction

Now, multiply both sides by \(4\) to eliminate the fraction:

\[ 9d = -36. \]

Step 6: Solve for \(d\)

Finally, divide by \(9\):

\[ d = -4. \]

Conclusion

Now that we have found the solution \(d = -4\), we can write a linear equation that represents this solution. A simple linear equation with the same solution set can be written as:

\[ d + 4 = 0. \]

This says that \(d\) is equal to \(-4\).

Thus, the linear equation that has the same solution set as

\[ \frac{9d}{4} + 3 - 2d = -2(3 + d) \]

is:

\[ \boxed{d + 4 = 0}. \]

these are the answer options
4w - 2(7 - 3w) = w/3 - 6
35 - 15b = 95
14 = 7/4 * x - 19/4 + 2x
(3x)/5 - 10 = -7

To find which of the provided answer options has the same solution set as the equation

\[ \frac{9d}{4} + 3 - 2d = -2(3 + d), \]

we first need to solve the original equation again to determine the solution.

Step 1: Simplify the Initial Equation

Starting from:

\[ \frac{9d}{4} + 3 - 2d = -2(3 + d), \]

we continue from prior steps:

  1. Rewrite the right side: \[ -2(3 + d) = -6 - 2d. \]

  2. Substitute: \[ \frac{9d}{4} + 3 - 2d = -6 - 2d. \]

  3. Simplify by eliminating \(-2d\) from both sides: \[ \frac{9d}{4} + 3 = -6. \]

  4. Subtract \(3\) from both sides: \[ \frac{9d}{4} = -9. \]

  5. Multiply by \(4\) to eliminate the fraction: \[ 9d = -36. \]

  6. Divide by \(9\): \[ d = -4. \]

Step 2: Rewrite the Solution

The solution we are looking for is \(d = -4\).

Step 3: Check Answer Options

Now, we will check each of the given equations to see which one results in \(d = -4\).

  1. Option 1: \[ 4w - 2(7 - 3w) = \frac{w}{3} - 6. \] Solving this: \[ 4w - 14 + 6w = \frac{w}{3} - 6, \] which simplifies to: \[ 10w - 14 = \frac{w}{3} - 6. \] Multiply through by \(3\) to eliminate the fraction: \[ 30w - 42 = w - 18. \] Rearranging: \[ 30w - w = -18 + 42 \implies 29w = 24 \implies w = \frac{24}{29}. \] This does not equal \(-4\).

  2. Option 2: \[ 35 - 15b = 95. \] Solving this: \[ -15b = 95 - 35 \implies -15b = 60 \implies b = -4. \] This equals \(-4\).

  3. Option 3: \[ 14 = \frac{7}{4}x - \frac{19}{4} + 2x. \] Convert \(2x\) to a fraction: \[ 14 = \frac{7}{4}x - \frac{19}{4} + \frac{8}{4}x \implies 14 = \frac{15}{4}x - \frac{19}{4}. \] Multiply through by \(4\) to eliminate the fraction: \[ 56 = 15x - 19. \] Rearranging gives: \[ 15x = 56 + 19 \implies 15x = 75 \implies x = 5. \] This does not equal \(-4\).

  4. Option 4: \[ \frac{3x}{5} - 10 = -7. \] Solving this: \[ \frac{3x}{5} = -7 + 10 \implies \frac{3x}{5} = 3. \] Multiply by \(5\): \[ 3x = 15 \implies x = 5. \] This does not equal \(-4\).

Conclusion

The only option that has the same solution set as the original equation is

\[ \boxed{35 - 15b = 95}. \]

This equation results in \(b = -4\), the same as the original solution we calculated.

Which option is the solution to the inequality -12g + 18 > -5g + 53?

To solve the inequality

\[ -12g + 18 > -5g + 53, \]

we'll isolate \(g\) step by step.

Step 1: Move all \(g\) terms to one side

  1. Start by adding \(12g\) to both sides to get all the \(g\) terms on one side: \[ 18 > -5g + 12g + 53. \]

  2. This simplifies to: \[ 18 > 7g + 53. \]

Step 2: Move constants to the other side

  1. Now, subtract \(53\) from both sides: \[ 18 - 53 > 7g. \]

  2. Simplifying gives: \[ -35 > 7g. \]

Step 3: Divide by 7

  1. Next, divide both sides by \(7\): \[ \frac{-35}{7} > g. \]

  2. Simplifying gives: \[ -5 > g, \] or equivalently, \[ g < -5. \]

Conclusion

The solution to the inequality is

\[ g < -5. \]

If you have specific options given for this inequality, please provide them, and I can confirm which one corresponds to \(g < -5\).

Graph x ≤ -3

To graph the inequality \(x \leq -3\), follow these steps:

Step 1: Draw the Number Line

  1. Draw a horizontal line representing the number line.
  2. Mark several key points on the line, making sure to include \(-3\). You might label points like \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), etc.

Step 2: Mark the Point \(-3\)

  1. At the point \(-3\), draw a solid dot (or closed circle). This indicates that \(-3\) is included in the solution set since the inequality includes \(\leq\).

Step 3: Shade the Appropriate Region

  1. Shade to the left of \(-3\) on the number line. This represents all values of \(x\) that are less than or equal to \(-3\).

Visual Representation

  • The graph will show a solid dot at \(-3\) and a shaded line extending to the left towards negative infinity:
<----|----|----|----|----|----|----|----|----|----|---->
    -5   -4   -3   -2   -1    0    1    2    3    4  
          ●==========================>

Here, the solid dot at \(-3\) means that \(-3\) is part of the solution set, and the shaded area to the left represents all numbers less than \(-3\).

Find the solution set of 4x - 3 > 9 and x + 6 > -2

To find the solution set for the compound inequalities \(4x - 3 > 9\) and \(x + 6 > -2\), we will solve each inequality step by step.

Step 1: Solve the first inequality \(4x - 3 > 9\)

  1. Start by adding \(3\) to both sides: \[ 4x > 9 + 3 \] \[ 4x > 12 \]

  2. Next, divide both sides by \(4\): \[ x > \frac{12}{4} \] \[ x > 3 \]

Step 2: Solve the second inequality \(x + 6 > -2\)

  1. Start by subtracting \(6\) from both sides: \[ x > -2 - 6 \] \[ x > -8 \]

Step 3: Combine the Results

We now have two inequalities:

  1. \(x > 3\)
  2. \(x > -8\)

The more restrictive inequality is \(x > 3\). Therefore, the solution set is:

\[ x > 3 \]

Step 4: Write the Solution Set

In interval notation, the solution set can be expressed as:

\[ (3, \infty) \]

Conclusion

Thus, the solution set for the compound inequalities \(4x - 3 > 9\) and \(x + 6 > -2\) is

\[ \boxed{(3, \infty)}. \]

these are the answer options
{x | x > 3}
The solution set is null.
{3, -8}
{x | x > -8}

From the previous calculations, we determined the solution set for the inequalities:

  1. \(4x - 3 > 9 \implies x > 3\)
  2. \(x + 6 > -2 \implies x > -8\)

Among these, the more restrictive solution is \(x > 3\).

Analyzing the Answer Options

Let's go through each of the options provided:

  1. \({ x | x > 3 }\): This correctly represents the solution set we found, which is all values greater than 3.
    (This is a valid option)

  2. The solution set is null.: This would suggest there are no solutions; however, we found valid solutions for \(x\).
    (This option is not valid)

  3. {3, -8}: This set lists specific values and does not include values greater than 3. Thus, it does not represent the solution set.
    (This option is not valid)

  4. \({ x | x > -8 }\): This option represents all values greater than \(-8\), which is broader than our solution set. While all values greater than 3 are also greater than -8, this option does not reflect the more restrictive condition we found.
    (This option is not valid)

Conclusion

The most accurate representation of the solution set is:

\[ \boxed{{ x | x > 3 }} \]

What is the undefined value for the equation 14 = (7x)/(9 + 4x)?

To find the undefined value for the equation

\[ 14 = \frac{7x}{9 + 4x}, \]

we need to determine when the denominator of the fraction is equal to zero because division by zero is undefined.

Step 1: Identify the Denominator

The denominator in the equation is:

\[ 9 + 4x. \]

Step 2: Set the Denominator to Zero

Set the denominator equal to zero to find the undefined values:

\[ 9 + 4x = 0. \]

Step 3: Solve for \(x\)

Now, solve for \(x\):

  1. Subtract \(9\) from both sides: \[ 4x = -9. \]

  2. Divide both sides by \(4\): \[ x = -\frac{9}{4}. \]

Conclusion

The value of \(x\) that makes the equation undefined is

\[ \boxed{-\frac{9}{4}}. \]

Solve for s in the following equation: 11/(s + 3) + 4 = 5

s = ?

To solve the equation

\[ \frac{11}{s + 3} + 4 = 5, \]

we will isolate the variable \(s\) step by step.

Step 1: Isolate the Fraction

Start by subtracting \(4\) from both sides:

\[ \frac{11}{s + 3} = 5 - 4. \]

This simplifies to:

\[ \frac{11}{s + 3} = 1. \]

Step 2: Eliminate the Fraction

Next, multiply both sides by \(s + 3\) to eliminate the fraction (assuming \(s + 3 \neq 0\)):

\[ 11 = 1(s + 3), \]

which simplifies to:

\[ 11 = s + 3. \]

Step 3: Solve for \(s\)

Now, subtract \(3\) from both sides:

\[ s = 11 - 3. \]

This gives:

\[ s = 8. \]

Conclusion

Thus, the solution for \(s\) in the equation

\[ \frac{11}{s + 3} + 4 = 5 \]

is

\[ \boxed{8}. \]

Which correctly shows the area formula for a trapezoid, A = 1/2 * h(b_{1} + b_{2}) rearranged for the quantity of interest h?

To rearrange the area formula for a trapezoid

\[ A = \frac{1}{2} \cdot h (b_1 + b_2), \]

for the height \(h\), we will go through the following steps:

Step 1: Multiply Both Sides by 2

To eliminate the \(\frac{1}{2}\), multiply both sides of the equation by \(2\):

\[ 2A = h(b_1 + b_2). \]

Step 2: Divide Both Sides by \((b_1 + b_2)\)

Now, to solve for \(h\), divide both sides by \((b_1 + b_2)\):

\[ h = \frac{2A}{b_1 + b_2}. \]

Conclusion

The rearranged formula for the height \(h\) in terms of the area \(A\) and the bases \(b_1\) and \(b_2\) is:

\[ \boxed{h = \frac{2A}{b_1 + b_2}}. \]

Given the formula for time in terms of distance and velocity, t = d/v, what operation would you use to isolate the distance?

To isolate the distance \(d\) in the formula for time in terms of distance and velocity, given by

\[ t = \frac{d}{v}, \]

you would perform the following operation:

Step: Multiply Both Sides by \(v\)

To isolate \(d\), multiply both sides of the equation by \(v\):

\[ t \cdot v = d. \]

Conclusion

Thus, the operation needed to isolate the distance \(d\) is multiplication by the velocity \(v\).

This yields the formula:

\[ d = t \cdot v. \]

So, you isolate distance by multiplying both sides by \(v\).

Sebastian needed to buy new tires for his car. The tire shop was running a holiday deal that allowed customers to buy three tires and get the fourth tire for $5. If the total price for all his tires was $237.50, what was the cost of each of the first three tires? (1 point)

Let the cost of each of the first three tires be represented as \( x \).

According to the problem:

  1. Sebastian buys three tires at the cost of \( x \) each.
  2. He buys a fourth tire for $5.

Thus, the total cost can be expressed as:

\[ 3x + 5 = 237.50 \]

Step 1: Subtract $5 from both sides

\[ 3x = 237.50 - 5 \]

Calculating the right side:

\[ 3x = 232.50 \]

Step 2: Divide both sides by 3

\[ x = \frac{232.50}{3} \]

Calculating \( x \):

\[ x = 77.50 \]

Conclusion

The cost of each of the first three tires is

\[ \boxed{77.50}. \]

A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.) (1 point)
1,280 + 30h = 0
30h = 10
1,280 - 30h = 0
10 - 30h = 0

To determine how long the water in the cooler will last, we need to establish an equation that relates the initial amount of water in ounces to the amount removed per hour.

Given Information:

  • The cooler holds 10 gallons of water.
  • \(10 \text{ gallons} = 1,280\) ounces (as given).
  • Each hour, 30 ounces of water are removed.

Setting Up the Equation:

Let \(h\) be the number of hours that the water lasts. After \(h\) hours, the total amount of water removed from the cooler is \(30h\) ounces.

Initially, the cooler has \(1,280\) ounces. Once all the water is gone (which is the point we need to find), we have this relationship:

\[ \text{Remaining water} = \text{Initial water} - \text{Water removed} \]

Thus, we can set it up as:

\[ 1,280 - 30h = 0. \]

This equation states that when the remaining water equals zero, the initial amount (1,280 ounces) minus the amount removed (30 ounces per hour multiplied by the number of hours) will equal zero.

Conclusion

The correct equation that shows how long the water in the cooler will last is:

\[ \boxed{1,280 - 30h = 0}. \]

Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2 * (4 - 8x) + 5x = 1 - 7x

(original and like a 9th wrote it)

To justify each step for solving the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) and determine the number of solutions, we’ll go through the process step-by-step, using the properties of equality.

Step 1: Distribute the \(\frac{3}{2}\)

Start with the original equation: \[ \frac{3}{2}(4 - 8x) + 5x = 1 - 7x. \]

Using the Distributive Property, we can distribute \(\frac{3}{2}\) to both terms inside the parentheses.

\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x. \]

Calculating those gives: \[ \frac{12}{2} - \frac{24}{2}x + 5x = 1 - 7x. \]

This simplifies to: \[ 6 - 12x + 5x = 1 - 7x. \]

Step 2: Combine Like Terms

Now combine the like terms on the left side: \[ 6 - 7x = 1 - 7x. \]

Step 3: Eliminate \(7x\) from Both Sides

Next, we can add \(7x\) to both sides to eliminate \(7x\) from the right side: \[ 6 = 1. \]

This immediately shows that we have obtained a statement that is always false. Specifically, \(6\) does not equal \(1\).

Conclusion: Determine Number of Solutions

The statement \(6 = 1\) indicates that the original equation has no solutions.

In conclusion:

  • The step of distributing the \(\frac{3}{2}\) and then combining like terms led us to a false statement, indicating there are no values of \(x\) that will satisfy the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x\).

Thus, the equation has:

\[ \boxed{\text{No solutions}}. \]

A car rental company charges $100 plus $0.25 per mile driven. Tyreek has a budget of $200 for his rental car. He wrote the equation 25x - 100 = 200 to figure out how many miles, x, he can drive in order to not go over his budget.
In 3 or 4 sentences, explain the error Tyreek made and then write a correct equation.
(original and like a 9th wrote it)

Tyreek made an error in his equation by incorrectly setting it up. He should have started with the total cost of the rental, which includes the fixed charge of $100 and the cost per mile of $0.25. Instead of writing \(25x - 100 = 200\), he should have expressed the total cost as \(100 + 0.25x = 200\), where \(x\) represents the number of miles driven.

The correct equation to determine how many miles he can drive without exceeding his budget is: \[ 100 + 0.25x = 200. \]

The electric current, / (amps), passing through an electronic component varies directly with the applied voltage, V (volts), according to the relationship V = IR where R, is the resistance measured in Ohms (Ω).
Wong is trying to find out the resistance, R, of an unknown component. He labels the resistance of the unknown component as Ω.
The resistance, R, of a circuit containing these unknown components is (5x + 20)Ω. Α 90-volt potential difference across the circuit produces a current of 2.25 amps.
What is the resistance of the unknown component? Explain how you arrived at your answer.

(original and like a 9th wrote it)

To find the resistance \( R \) of the unknown component, we can use Ohm's law, which states the relationship between voltage (\( V \)), current (\( I \)), and resistance (\( R \)) as:

\[ V = IR. \]

We know from the problem that:

  • The applied voltage \( V \) is 90 volts.
  • The current \( I \) is 2.25 amps.
  • The resistance \( R \) is given by the expression \( (5x + 20) , \Omega \).

Step 1: Apply Ohm's Law

Using Ohm's law, we can express the resistance as follows:

\[ R = \frac{V}{I}. \]

Step 2: Substitute the Values

Now, substitute the values of \( V \) and \( I \):

\[ R = \frac{90}{2.25}. \]

Step 3: Calculate the Resistance

Now, perform the division:

\[ R = 40 , \Omega. \]

Step 4: Set the Resistance Expression Equal to the Calculated Resistance

Since we know that the resistance is also expressed as \( R = (5x + 20) , \Omega \), we can set up the equation:

\[ 5x + 20 = 40. \]

Step 5: Solve for \( x \)

Now, we can solve for \( x \):

  1. Subtract 20 from both sides: \[ 5x = 40 - 20, \] \[ 5x = 20. \]

  2. Divide both sides by 5: \[ x = \frac{20}{5} = 4. \]

Conclusion

Thus, the value of \( x \) is 4. This means that the resistance of the unknown component is \( R = (5x + 20) \Omega = (5(4) + 20) \Omega = (20 + 20) \Omega = 40 \Omega \).

So, the resistance of the unknown component is \( \boxed{40 , \Omega} \), because we used Ohm's law to find the resistance from the voltage and current provided.

Using the properties of inequality, justify both steps in solving the inequality 2/5 * m - 7 ≤ 17
(original and like a 9th wrote it)

To solve the inequality

\[ \frac{2}{5}m - 7 \leq 17, \]

we will go through the steps one by one, using the properties of inequalities for justification.

Step 1: Add 7 to Both Sides

Original Inequality: \[ \frac{2}{5}m - 7 \leq 17. \]

Justification: We want to isolate the term containing \( m \). We do this by adding \( 7 \) to both sides of the inequality. This is known as the Addition Property of Inequality, which states that if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \).

Performing the Operation: \[ \frac{2}{5}m - 7 + 7 \leq 17 + 7. \] This simplifies to: \[ \frac{2}{5}m \leq 24. \]

Step 2: Multiply Both Sides by \(\frac{5}{2}\)

Current Inequality: \[ \frac{2}{5}m \leq 24. \]

Justification: To solve for \( m \), we need to eliminate the coefficient \(\frac{2}{5}\). We can do this by multiplying both sides of the inequality by the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\). This is allowed because multiplying by a positive number preserves the direction of the inequality.

Performing the Operation: \[ \frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2}. \] Now simplify: \[ m \leq 24 \cdot \frac{5}{2}. \] Calculating the right side: \[ 24 \cdot \frac{5}{2} = \frac{120}{2} = 60. \]

Conclusion

The solution to the inequality is: \[ m \leq 60. \]

To summarize the justification:

  1. We added \(7\) to both sides to isolate the term with \( m \), which is justified by the Addition Property of Inequalities.
  2. We then multiplied both sides by \(\frac{5}{2}\) to solve for \( m\), which is justified by the fact that multiplying both sides of an inequality by a positive number preserves the inequality's direction.

Thus, the final result is:

\[ \boxed{m \leq 60}. \]

Rewrite the equation (4x)/(3x - 12) + 3 = 10 in two equations joined by "and."
(original and like a 9th wrote it)

To rewrite the equation

\[ \frac{4x}{3x - 12} + 3 = 10 \]

in two separate equations joined by "and," we can break it down into two parts.

Step 1: Isolate the Fraction

First, let's isolate the fraction by moving \(3\) to the other side of the equation.

This gives us:

\[ \frac{4x}{3x - 12} = 10 - 3 \]

which simplifies to:

\[ \frac{4x}{3x - 12} = 7. \]

Step 2: Rewrite as Two Equations

Now we can rewrite the original equation as two equations joined by "and." The two equations will represent the condition of the fraction being equal to \(7\) and the condition we derived from the original equation that \(3x - 12\) should not equal \(0\) (to ensure the denominator is not zero).

  1. \(\frac{4x}{3x - 12} = 7\) (because that's what we derived when isolating the fraction).
  2. \(3x - 12 \neq 0\) (to ensure that we don’t divide by zero).

Thus, we can write:

\[ \frac{4x}{3x - 12} = 7 \quad \text{and} \quad 3x - 12 \neq 0. \]

Final Answer

So the equation rewritten in two equations joined by "and" is:

\[ \boxed{\frac{4x}{3x - 12} = 7 \quad \text{and} \quad 3x - 12 \neq 0.} \]

Rearrange the slope formula y = mx + b for the quantity of interest m. Make sure to show each step.
(original and like a 9th wrote it)

The slope formula is given by

\[ y = mx + b, \]

where \(y\) is the dependent variable, \(m\) is the slope, \(x\) is the independent variable, and \(b\) is the y-intercept.

To rearrange this equation for \(m\), we'll follow these steps:

Step 1: Isolate the term containing \(m\)

First, we want to get rid of the \(b\) on the right side. We do this by subtracting \(b\) from both sides:

\[ y - b = mx. \]

Step 2: Solve for \(m\)

Now we want to isolate \(m\). Since \(m\) is currently multiplied by \(x\), we can divide both sides by \(x\) (assuming \(x \neq 0\)):

\[ m = \frac{y - b}{x}. \]

Conclusion

Now we have expressed \(m\) in terms of \(y\), \(b\), and \(x\). The rearranged formula for the slope \(m\) is:

\[ \boxed{m = \frac{y - b}{x}}. \]

Dezeree received a $100 gift card and needs new sweaters for school. The sweaters cost $27.50 each. Write an equation to determine how many sweaters she can buy and spend exactly $100, solve it, and determine if the value is a viable solution. (4 points)
(original and like a 9th wrote it)

To determine how many sweaters Dezeree can buy with her $100 gift card, we can set up an equation based on the cost of each sweater and the total amount she can spend.

Step 1: Write the Equation

Let \( x \) be the number of sweaters Dezeree can buy. Since each sweater costs $27.50, the total cost for the sweaters is \( 27.50x \).

The equation to reflect the total amount spent should equal the gift card amount of $100:

\[ 27.50x = 100. \]

Step 2: Solve the Equation

Now we will solve for \( x \):

  1. Divide both sides of the equation by 27.50 to isolate \( x \):

\[ x = \frac{100}{27.50}. \]

  1. Performing the division:

\[ x \approx 3.636. \]

Step 3: Determine if the Value is a Viable Solution

Since \( x \) must be a whole number (you can't buy a fraction of a sweater), we need to round \( x \) to the nearest whole number. In this case, Dezeree can buy either:

  • 3 sweaters:

    \[ 27.50 \times 3 = 82.50 \quad \text{(total spent)}. \]

  • 4 sweaters:

    \[ 27.50 \times 4 = 110 \quad \text{(total spent)}. \]

Buying 4 sweaters would exceed her $100 limit, which means she can't afford it. When Dezeree buys 3 sweaters, she spends $82.50, which leaves her with $17.50 remaining.

Conclusion

Dezeree can buy a maximum of 3 sweaters with her $100 gift card. Therefore, the solution \( x = 3 \) is a viable solution because it does not exceed her budget.

\[ \boxed{3 \text{ sweaters}} \]