Question

Alisha solved the inequality y ≤ |3x − 6| + 2 and got y ≤ 3x − 4 for x ≥ 2 and y ≤ −3x − 4 for x < 2. Explain Alisha's error, and show the correct answer.

Answer 1

Given:

The inequality is
`y \leq |3x - 6| + 2`.

Alisha solved the given inequality and got

`y \leq 3x -4` for
`x \geq 2` and
`y \leq -3x - 4` for
`x < 2`.

To find:

The Alisha's error, and the correct answer.

Solution:

We have,

`y\leq |3x-6|+2`

For
`x \geq 2, |3x-6|=3x-6`. So,

`y\leq 3x-6+2`

`y\leq 3x-4`

For
`x < 2, |3x-6|=-(3x-6)`. So,

`y\leq -(3x-6)+2`

`y\leq -3x+6+2`

`y\leq -3x+8`

Alisha's second inequality is wrong because she did not distribute the negative with 6.

Therefore, the correct answer is

`y \leq 3x -4` for
`x \geq 2` and
`y \leq -3x +8` for
`x < 2`.