Question

15. Match the inequality to its graphed solution.(2 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.

Numberlines: 1st numberline shows -2 going to the left and is open, the 2nd numberline shows +12 going to the left
Answer choices:8x < 4x + 24, -3 ( x-12) > 0, -5 (x+5) > - 15, 1/4 (8x+16) > -4

Answer 1

Solve the equation filling in the steps: 2(x−3)+21=−3 (6 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. Distribute to get Response area Combine like terms to get Response area Use the (choose a property) Response area to get (resulting equation) Response area Use the (choose a property) Response area to get (resulting answer) Response area

Answer 2

18. Solve the following:(2 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. ax−44<15 ax+15=44

Answer 3

To match the inequalities to their graphed solutions based on the provided information about the number lines:

1. **First Number Line**: This number line shows -2 going to the left, and the endpoint is open. This indicates that the inequality is of the form \( x < -2 \) or \( x > -2 \).

2. **Second Number Line**: This number line shows +12 going to the left, suggesting that it is an inequality of the form \( x < 12 \).

Let's analyze the given inequalities:

1. **\( 8x < 4x + 24 \)**: Simplifying this, we get \( 4x < 24 \) or \( x < 6 \).
2. **\( -3(x - 12) > 0 \)**: Simplifying this, we find \( -3x + 36 > 0 \) or \( -3x > -36 \), which gives \( x < 12 \).
3. **\( -5(x + 5) > -15 \)**: Simplifying gives \( -5x - 25 > -15 \), which leads to \( -5x > 10 \) or \( x < -2 \).
4. **\( \frac{1}{4}(8x + 16) > -4 \)**: Solving this results in \( 2x + 4 > -4 \) which simplifies to \( 2x > -8 \) or \( x > -4 \).

**Matching the inequalities:**
- The first number line (open circle at -2, going left) corresponds to **\( -5(x+5) > -15 \)** since it leads to \( x < -2 \).
- The second number line (open circle at +12, going left) corresponds to **\( -3(x - 12) > 0 \)** as that results in \( x < 12 \).

**Final Answers:**
1. First number line: **-5(x + 5) > -15**
2. Second number line: **-3(x - 12) > 0**

Make the responses match as indicated.

Answer 4

Let's break down the solution to the equation \( 2(x - 3) + 21 = -3\), filling in the necessary steps.

1. **Distribute to get**:
- Distributing \(2\) into \((x - 3)\):
\[
2 \cdot x - 2 \cdot 3 = 2x - 6
\]
So, we have:
\[
2x - 6 + 21 = -3
\]

2. **Combine like terms to get**:
- Combine \(-6\) and \(21\):
\[
2x + 15 = -3
\]

3. **Use the (choose a property)**:
- Use the **addition property of equality** to subtract \(15\) from both sides:
\[
2x + 15 - 15 = -3 - 15
\]
Resulting in:
\[
2x = -18
\]

4. **Use the (choose a property)**:
- Use the **division property of equality** to divide both sides by \(2\):
\[
\frac{2x}{2} = \frac{-18}{2}
\]
Resulting in:
\[
x = -9
\]

### Summary of Steps:
- Distribute to get: **\(2x - 6 + 21 = -3\)**
- Combine like terms to get: **\(2x + 15 = -3\)**
- Use the **addition property of equality** to get: **\(2x = -18\)**
- Use the **division property of equality** to get: **\(x = -9\)**

You can fill in each response area with the corresponding explanation above.

Answer 5

To solve the given equations and inequalities step-by-step, let's break them down.

### Solve the inequality: \( ax - 44 < 15 \)

1. **Isolate the variable term:**
To isolate the term \( ax \), add \( 44 \) to both sides:
\[
ax < 15 + 44
\]
Resulting in:
\[
ax < 59
\]

2. **Solve for \( x \):**
Divide both sides by \( a \) (assuming \( a \neq 0 \)):
\[
x < \frac{59}{a}
\]

### Solve the equation: \( ax + 15 = 44 \)

1. **Isolate the variable term:**
To isolate \( ax \), subtract \( 15 \) from both sides:
\[
ax = 44 - 15
\]
Resulting in:
\[
ax = 29
\]

2. **Solve for \( x \):**
Divide both sides by \( a \) (assuming \( a \neq 0 \)):
\[
x = \frac{29}{a}
\]

### Final Results:

1. For the inequality \( ax - 44 < 15 \):
- Result: **\( x < \frac{59}{a} \)**

2. For the equation \( ax + 15 = 44 \):
- Result: **\( x = \frac{29}{a} \)**

Fill in the response areas accordingly with the above simplifications.