Question

ax-4<12

Answer 1

this equation involves a numberline

Answer 2

To solve the inequality \( ax - 4 < 12 \), you want to isolate \( x \). Here are the steps:

1. Add 4 to both sides of the inequality:
\[
ax - 4 + 4 < 12 + 4
\]
This simplifies to:
\[
ax < 16
\]

2. Now, divide both sides by \( a \), keeping in mind that the direction of the inequality will change if \( a \) is negative:
\[
x < \frac{16}{a} \quad \text{if } a > 0
\]
or
\[
x > \frac{16}{a} \quad \text{if } a < 0
\]

Thus, the solution to the inequality depends on the value of \( a \):

- If \( a > 0 \), then the solution is \( x < \frac{16}{a} \).
- If \( a < 0 \), then the solution is \( x > \frac{16}{a} \).
- If \( a = 0 \), the inequality becomes \( -4 < 12 \), which is always true, meaning \( x \) can be any value.

Please let me know if you need further assistance!

Answer 3

To represent the solution of the inequality \( ax - 4 < 12 \) on a number line, we first need to determine the specific case based on the value of \( a \). Let’s outline the steps for each potential case and then describe how you would represent it on a number line.

### Case 1: \( a > 0 \)

1. Solve the inequality:
\[
x < \frac{16}{a}
\]

2. **Number Line Representation**:
- Draw a number line.
- Find the point \( \frac{16}{a} \) on the number line.
- Since \( x < \frac{16}{a} \), you will place an open circle at \( \frac{16}{a} \) (indicating that this point is not included in the solution).
- Shade to the left of \( \frac{16}{a} \) to indicate all values less than \( \frac{16}{a} \).

### Case 2: \( a < 0 \)

1. Solve the inequality:
\[
x > \frac{16}{a}
\]

2. **Number Line Representation**:
- Draw a number line.
- Find the point \( \frac{16}{a} \) on the number line.
- Since \( x > \frac{16}{a} \), you will place an open circle at \( \frac{16}{a} \).
- Shade to the right of \( \frac{16}{a} \) to indicate all values greater than \( \frac{16}{a} \).

### Case 3: \( a = 0 \)

- The original inequality simplifies to \( -4 < 12 \), which is always true for any value of \( x \).
- **Number Line Representation**:
- The entire number line is shaded, indicating that all real numbers are included in the solution.

### Summary of Number Line Representations:
- **For \( a > 0 \)**: Open circle at \( \frac{16}{a} \) with shading to the left.
- **For \( a < 0 \)**: Open circle at \( \frac{16}{a} \) with shading to the right.
- **For \( a = 0 \)**: Entire number line shaded.

If you can provide a specific value for \( a \), I can help you create a more precise representation!