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 $x < 2$.