Which of the x values are solutions to the inequality 4(2 – x) > –2x – 3(4x + 1)? Check all that apply.

Question

x = –1.1
x = –2.2
x = 0
x = –10
x = 10

Answer 1

To solve the inequality \( 4(2 - x) > -2x - 3(4x + 1) \), let's start by simplifying both sides of the inequality.

1. Distribute the \( 4 \) on the left side: \[ 4(2 - x) = 8 - 4x \]

2. Distribute the \( -3 \) on the right side: \[ -3(4x + 1) = -12x - 3 \] So, the right side becomes: \[ -2x - 12x - 3 = -14x - 3 \]

Thus, the inequality simplifies to: \[ 8 - 4x > -14x - 3 \]

3. Next, let's add \( 14x \) to both sides: \[ 8 - 4x + 14x > -3 \] Simplifying gives: \[ 8 + 10x > -3 \]

4. Now, subtract \( 8 \) from both sides: \[ 10x > -3 - 8 \] So, we have: \[ 10x > -11 \]

5. Finally, divide by \( 10 \): \[ x > -\frac{11}{10} \] or \[ x > -1.1 \]

Now that we have the solution \( x > -1.1 \), we can check the given x values:

• \( x = -1.1 \): Not a solution (since it is not greater than -1.1)
• \( x = -2.2 \): Not a solution (since -2.2 is less than -1.1)
• \( x = 0 \): Solution (since 0 is greater than -1.1)
• \( x = -10 \): Not a solution (since -10 is less than -1.1)
• \( x = 10 \): Solution (since 10 is greater than -1.1)

Thus, the values that are solutions to the inequality are:

• x = 0
• x = 10

Answer 2

The inequality given is ( \frac{3}{4}(x + 8) > \frac{1}{2}(x + 10) ). You need to solve this inequality and then graph the solution on the number line provided, which ranges from -5 to 5. open or close circle? left or right

Answer 3

To solve the inequality

\[ \frac{3}{4}(x + 8) > \frac{1}{2}(x + 10), \]

let's start by clearing the fractions.

1. Multiply both sides by 4 to eliminate the denominators: \[ 4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10). \]

   This simplifies to: \[ 3(x + 8) > 2(x + 10). \]

2. Distribute on both sides: \[ 3x + 24 > 2x + 20. \]

3. Subtract \(2x\) from both sides: \[ 3x - 2x + 24 > 20, \]

   simplifying to: \[ x + 24 > 20. \]

4. Now, subtract \(24\) from both sides: \[ x > 20 - 24, \]

   which simplifies to: \[ x > -4. \]

Graphing the Solution

The solution \(x > -4\) means that \(x\) can take any value greater than \(-4\), but not equal to \(-4\).

1. Open Circle: Since \(-4\) is not included in the solution (the inequality is strict), you would use an open circle at \(-4\).
2. Arrow to the Right: This indicates that the solution continues to the right (greater than \(-4\)).

Number Line

On a number line ranging from \(-5\) to \(5\):

• Place an open circle at \(-4\).
• Shade or draw an arrow to the right from \(-4\) to indicate all values greater than \(-4\).

Here's a representation in text form:

---|----|----|----|----|----|----|----
  -5   -4   -3   -2   -1    0    1
        (====================>

In conclusion, the solution is \(x > -4\) with an open circle at \(-4\) and shading to the right.

Answer 4

incorrect

Answer 5

Let's go through the inequality step by step again to ensure we get it correct:

The original inequality is: \[ \frac{3}{4}(x + 8) > \frac{1}{2}(x + 10). \]

Step 1: Clear the fractions

Multiply both sides by 4 (the least common multiple of the denominators) to eliminate the fractions: \[ 4 \times \frac{3}{4}(x + 8) > 4 \times \frac{1}{2}(x + 10). \] This simplifies to: \[ 3(x + 8) > 2(x + 10). \]

Step 2: Distribute

Distributing on both sides gives us: \[ 3x + 24 > 2x + 20. \]

Step 3: Move all terms involving \(x\) to one side

Subtract \(2x\) from both sides: \[ 3x - 2x + 24 > 20. \] This simplifies to: \[ x + 24 > 20. \]

Step 4: Isolate \(x\)

Subtract 24 from both sides: \[ x > 20 - 24, \] which simplifies to: \[ x > -4. \]

Conclusion of the solution

The solution to the inequality is \(x > -4\).

Graph on the number line

• Open Circle at \(-4\): indicates that \(-4\) is not included in the solution.
• Arrow to the Right: indicates that the solution includes all numbers greater than \(-4\).

In a basic number line representation between \(-5\) and \(5\):

---|----|----|----|----|----|----|----
  -5   -4   -3   -2   -1    0    1
        (====================>

Summary

• Open circle at \(-4\) (because \(-4\) is not part of the solution)
• Shade or draw an arrow to the right indicating all values greater than \(-4\).

If you still consider this incorrect, please clarify which particular part seems off, and I’ll do my best to address it!

Answer 6

its genuinely uncorrect i put open circle arrow going right on -4 and its wrong?

Answer 7

I apologize for the confusion earlier. Let's re-evaluate the inequality clearly step-by-step and ensure that we understand the graphing part correctly.

The inequality we are solving is:

\[ \frac{3}{4}(x + 8) > \frac{1}{2}(x + 10). \]

Step 1: Clear the fractions

To eliminate the fractions, we can multiply both sides by 4: \[ 4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(x + 10). \]

This simplifies to: \[ 3(x + 8) > 2(x + 10). \]

Step 2: Distribute

Now distribute on both sides: \[ 3x + 24 > 2x + 20. \]

Step 3: Isolate \(x\)

Subtract \(2x\) from both sides: \[ 3x - 2x + 24 > 20 \implies x + 24 > 20. \]

Now subtract 24 from both sides: \[ x > 20 - 24 \implies x > -4. \]

Interpretation

The solution is \(x > -4\).

Graphing the Solution

1. Open Circle at -4: Since the inequality is strict (greater than, not greater than or equal to), we use an open circle at \(-4\).
2. Arrow to the Right: This indicates that values greater than \(-4\) are included in the solution.

Number Line Representation

On a number line from -5 to 5:

• Draw an open circle at \(-4\).
• Shade or draw an arrow pointing right from \(-4\) to indicate all values greater than \(-4\).

Here’s how it looks:

---|----|----|----|----|----|----|----
  -5   -4   -3   -2   -1    0    1
        O====================>

The O indicates the open circle at \(-4\).

Conclusion

• The solution \(x > -4\) is correct.
• You should indeed use an open circle at \(-4\) and shade to the right.

If there is still a misunderstanding or a specific aspect of the graph that seems incorrect, please let me know! Thank you for your patience.

Answer 8

You need to solve this inequality and then graph the solution on the number line provided, which ranges from -5 to 5. its 3/4(x+8)>1/2(2x+10) open or closed circle? left or right

Answer 9

Thank you for your patience, and I appreciate the opportunity to clarify everything. Let’s solve the correct inequality you just provided:

\[ \frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10). \]

Step 1: Clear the fractions

Multiply both sides by 4 to eliminate the denominators:

\[ 4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10). \]

This simplifies to:

\[ 3(x + 8) > 2(2x + 10). \]

Step 2: Distribute both sides

Distributing on both sides gives:

\[ 3x + 24 > 4x + 20. \]

Step 3: Isolate \(x\)

Now, we'll isolate \(x\) by moving all terms involving \(x\) to one side and constant terms to the other side:

1. Subtract \(3x\) from both sides:

\[ 24 > x + 20. \]

2. Next, subtract \(20\) from both sides:

\[ 24 - 20 > x \implies 4 > x. \]

This can also be written as:

\[ x < 4. \]

Summary of the Solution

The solution to the inequality is:

\[ x < 4. \]

Step 4: Graphing on the number line

• Open Circle at \(4\): Since the inequality is strict (<), you would place an open circle at \(4\).
• Arrow to the Left: This indicates that \(x\) can take any value less than \(4\).

Number Line Representation

On a number line that ranges from -5 to 5, it would look like this:

---|----|----|----|----|----|----|----
 -5   -4   -3   -2   -1    0    1    2    3    4    5
        O<===================

Where O represents the open circle at \(4\) and the shading to the left represents all values less than \(4\).

Conclusion

• Use an open circle at \(4\).
• Shade or draw an arrow to the left indicating all values less than \(4\).

Thank you for bringing this to my attention. If you have any further questions, feel free to ask!

Answer 10

no no my mistake im sorry thank you tho

What is the solution to 3/4(x+8)>1/2(2x+10)?

(–∞, –4)
(–4, ∞)
(–∞, 4)
(4, ∞)

Answer 11

No problem at all! Let's solve the inequality step by step to determine its solution.

Given the inequality:

\[ \frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10). \]

Step 1: Clear the fractions

Multiply both sides by 4 to eliminate the fractions:

\[ 4 \cdot \frac{3}{4}(x + 8) > 4 \cdot \frac{1}{2}(2x + 10). \]

This simplifies to:

\[ 3(x + 8) > 2(2x + 10). \]

Step 2: Distribute both sides

Now distribute on both sides:

\[ 3x + 24 > 4x + 20. \]

Step 3: Isolate \(x\)

Now we'll move terms involving \(x\) to one side and constant terms to the other side:

1. Subtract \(3x\) from both sides:

\[ 24 > x + 20. \]

2. Subtract \(20\) from both sides:

\[ 24 - 20 > x \implies 4 > x. \]

This can also be expressed as:

\[ x < 4. \]

Final Solution

The solution is \(x < 4\), which operates over the interval:

\[ (-\infty, 4). \]

Answer

Thus, among the options provided, the correct answer is:

\[ \text{(–∞, 4)}. \]

If you have any further questions or need additional clarification, feel free to ask!