Question

Which are correct representations of the inequality 6x ≥ 3 + 4(2x – 1)? Select three options.

1 ≥ 2x
6x ≥ 3 + 8x – 4
A number line from negative 1.5 to 1.5 in increments of 0.5. A point is at 0.5 and a bold line starts at 0.5 and is pointing to the left.
A number line from negative 1.5 to 1.5 in increments of 0.5. A point is at negative 0.5 and a bold line starts at negative 0.5 and is pointing to the right.
A number line from negative 1.5 to 1.5 in increments of 0.5. A point is at 0.5 and a bold line starts at 0.5 and is pointing to the right.

Answer 1

To find the correct representations of the inequality \( 6x \geq 3 + 4(2x - 1) \), we should first simplify the inequality.

1. Start with the original inequality:
\[
6x \geq 3 + 4(2x - 1)
\]

2. Distribute \( 4 \) inside the parentheses:
\[
6x \geq 3 + 8x - 4
\]

3. Combine the constants on the right side:
\[
6x \geq 8x - 1
\]

4. Rearranging the inequality (subtract \( 8x \) from both sides):
\[
6x - 8x \geq -1
\]
This simplifies to:
\[
-2x \geq -1
\]

5. Dividing both sides by -2 (remember to flip the inequality sign):
\[
x \leq \frac{1}{2}
\]

Now, let's evaluate the options for correctness:

- **Option 1:** \( 1 \geq 2x \)
- This can be rearranged to \( x \leq \frac{1}{2} \). This option is correct.

- **Option 2:** \( 6x \geq 3 + 8x - 4 \)
- This is the same form we derived when we simplified the original inequality. This option is correct.

- **Option 3:** A number line from -1.5 to 1.5, a point at 0.5, and a bold line pointing to the left.
- This indicates \( x \leq 0.5 \), which is a correct representation.

- **Option 4:** A number line from -1.5 to 1.5, a point at -0.5, and a bold line pointing to the right.
- This indicates \( x \geq -0.5 \), which is incorrect based on our derived inequality.

- **Option 5:** A number line from -1.5 to 1.5, a point at 0.5, and a bold line pointing to the right.
- This indicates \( x \geq 0.5 \), which is also incorrect based on our derived inequality.

Based on the analysis, the three correct representations are:

1. \( 1 \geq 2x \)
2. \( 6x \geq 3 + 8x - 4 \)
3. The number line with a point at 0.5 and a bold line pointing to the left (indicating \( x \leq 0.5 \)).