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Using the Distributive Property as a good first step to solving the equation 7(5x+2)=−4(6−5x), you could simplify this equation to get which of these choices?(1 point)

Responses

35x+14=−24−20x
35 x plus 14 is equal to negative 24 minus 20 x

35x+2=−24−5x
35 x plus 2 is equal to negative 24 minus 5 x

35x+14=−24+20x
35 x plus 14 is equal to negative 24 plus 20 x

35x+2=24−20x
35 x plus 2 is equal to 24 minus 20 x

5x+14=6+20x

11 answers

To simplify the equation \( 7(5x + 2) = -4(6 - 5x) \) using the Distributive Property, we first apply the property to both sides:

  1. On the left side: \[ 7(5x + 2) = 7 \cdot 5x + 7 \cdot 2 = 35x + 14 \]

  2. On the right side: \[ -4(6 - 5x) = -4 \cdot 6 + (-4)(-5x) = -24 + 20x \]

Now we can set these results equal to one another: \[ 35x + 14 = -24 + 20x \]

Therefore, the simplified equation we obtain from using the Distributive Property is: \[ 35x + 14 = -24 + 20x \]

The correct choice among the provided responses is: \[ 35x + 14 = -24 + 20x \]

For the equation 2(x−5)=9−3x+6+8+3x+7, the left side of the equation can be simplified using the Distributive Property, and the right side of the equation can be simplified by combining like terms. (2 points)
Simplify the left side of the equation: [Response area]

Simplify the right side of the equation: [response area]

responses
2x - 10
36x
2x - 5
6x + 30
30
30x

Let's start simplifying each side of the equation \( 2(x - 5) = 9 - 3x + 6 + 8 + 3x + 7 \).

Left Side:

Using the Distributive Property, we simplify the left side: \[ 2(x - 5) = 2 \cdot x - 2 \cdot 5 = 2x - 10 \]

Right Side:

Next, we combine like terms on the right side: \[ 9 - 3x + 6 + 8 + 3x + 7 \] Combine the constant terms: \[ 9 + 6 + 8 + 7 = 30 \] And combine the \(x\) terms: \[ -3x + 3x = 0 \] Thus, the right side simplifies to: \[ 0 + 30 = 30 \]

Summary of Simplifications:

  • Left Side: \(2(x - 5) \rightarrow 2x - 10\)
  • Right Side: \(9 - 3x + 6 + 8 + 3x + 7 \rightarrow 30\)

So, the responses are:

  • Left side: \(2x - 10\)
  • Right side: \(30\)
Jill solves the equation 3(x−5)+2x=0
. Fill in the blanks for each step with the correct values:(3 points)
3x+[Response area]+2x=0
5x=[Response area]
x=[Response area]

responses
-3
3
15
1
-15
5
-5
-1

Let's solve the equation step-by-step and fill in the blanks as we go.

Original Equation:

\[ 3(x - 5) + 2x = 0 \]

Step 1: Use the Distributive Property

Distributing \(3\) to both \(x\) and \(-5\): \[ 3x - 15 + 2x = 0 \] This means the first blank should be \(-15\).

Step 2: Combine Like Terms

Now, combine the \(x\) terms: \[ (3x + 2x) - 15 = 0 \] This simplifies to: \[ 5x - 15 = 0 \] So the second blank should be \(-15\).

Step 3: Add \(15\) to Both Sides

To isolate \(5x\): \[ 5x = 15 \] So the third blank should be \(15\).

Step 4: Solve for \(x\)

Divide both sides by \(5\): \[ x = \frac{15}{5} = 3 \] So the final answer for \(x\) is \(3\).

Summary of the Responses:

  1. The first blank: \(-15\)
  2. The second blank: \(15\)
  3. The third blank: \(3\)

Therefore, the completed steps are:

  • \(3x - 15 + 2x = 0\)
  • \(5x = 15\)
  • \(x = 3\)
After combining like terms to simplify the equation 3−15x+24+16x=4x−24−4x
, what would be the next best step to finish solving?(1 point)
Responses

Subtract 27 from both sides of the equation.
Subtract 27 from both sides of the equation.

Add x to both sides of the equation.
Add x to both sides of the equation.

Divide both sides of the equation by 15.
Divide both sides of the equation by 15.

Subtract 24 from both sides of the equation.
Subtract 24 from both sides of the equation.

Subtract x from both sides of the equation.

Let's first combine like terms on both sides of the equation \(3 - 15x + 24 + 16x = 4x - 24 - 4x\).

Simplifying the left side:

  • Combine constants: \(3 + 24 = 27\)
  • Combine \(x\) terms: \(-15x + 16x = 1x\)

So, the left side simplifies to: \[ 27 + 1x \quad \text{or} \quad x + 27 \]

Simplifying the right side:

  • Combine like terms: \(4x - 4x = 0\)
  • Thus, the right side simplifies to: \[ -24 \]

Complete simplified equation:

Now we have: \[ x + 27 = -24 \]

Next Step:

The next best step to isolate \(x\) would be to subtract \(27\) from both sides of the equation:

\[ x + 27 - 27 = -24 - 27 \] This will give: \[ x = -24 - 27 = -51 \]

So, the correct response for the next step is: Subtract 27 from both sides of the equation.

Which inequality represents the graph below:

(1 point)
Responses

3x−7≤−10
3 x minus 7 is less than or equal to negative 10

−3x−7≥−10
negative 3 x minus 7 is greater than or equal to negative 10

3x+7>10
3 x plus 7 is greater than 10

−3x+7<10

To solve for the inequality that represents the graph, let's analyze the possible inequalities given the standard forms:

  1. \(3x - 7 \leq -10\)

    • Add 7 to both sides: \[ 3x \leq -3 \quad \text{(divide both sides by 3)} \quad x \leq -1 \]

  2. \(-3x - 7 \geq -10\)

    • Add 7 to both sides: \[ -3x \geq -3 \quad \text{(dividing by -3 reverses the inequality)} \quad x \leq 1 \]

  3. \(3x + 7 > 10\)

    • Subtract 7 from both sides: \[ 3x > 3 \quad \text{(divide both sides by 3)} \quad x > 1 \]

  4. \(-3x + 7 < 10\)

    • Subtract 7 from both sides: \[ -3x < 3 \quad \text{(dividing by -3 reverses the inequality)} \quad x > -1 \]

To determine which inequality corresponds with the graph, you would typically look for key information such as the direction of the inequality (<, ≤, >, ≥) and the critical value (the threshold where the variable x is evaluated).

If you have more context regarding the graph (such as the number line or the region shaded), please provide that information. Based on these inequalities alone, we would determine the correct option based on direction of the shaded area and specific values indicated in the graph.

If you were to identify a specific one given a conventional number line graph, typical choices would be:

  • If the graph includes a point marked with a filled circle (inclusive), it would suggest either the options involving "≤" or "≥".
  • If it includes an open circle (exclusive), the options would include either "<" or ">".

Please specify how the graph appears if possible, or if you have any of that context available, let me know!

he speed limit sign on the expressway says that the speed limit is 55 mph. Write an inequality that represents all speeds you can legally drive on the expressway. (1 point)
Responses

x>55
x is greater than 55

x≤55
x is less than or equal to 55

x≥55
x is greater than or equal to 55

x<55

The speed limit sign indicates that the legal maximum speed you can drive on the expressway is 55 mph. This means you can drive at a speed less than or equal to 55 mph.

Therefore, the appropriate inequality that represents all speeds you can legally drive on the expressway is:

\( x \leq 55 \)

So the correct response is:

  • \(x \leq 55\) (x is less than or equal to 55)