Question

Which best describes the solution set of the compound inequality below?

2 + x ≤ 3x – 6 ≤ 12

4 ≤ x ≤ 9
4 ≤ x ≤ 6
–2 ≤ x ≤ 2
–2 ≤ x ≤ 3

Answer 1

4 ≤ x ≤ 6

Explanation:

To solve the compound inequality, solve each part of the inequality separately and then find the values of x that satisfy both inequalities.

Solve the left inequality:

`\begin{aligned}2 + x &\leq 3x - 6\\\\2 + x -x &\leq 3x - 6 -x\\\\2&\leq 2x - 6\\\\2+6&\leq 2x - 6+6\\\\8&\leq 2x\\\\2x &\geq 8\\\\(2x)/(2) &\geq (8)/(2)\\\\x &\geq 4\end{aligned}`

So, the solution to the left inequality is x ≥ 4.

Solve the right inequality:

`\begin{aligned}3x-6&\leq 12\\\\3x-6+6&\leq 12+6\\\\3x&\leq 18\\\\(3x)/(3)&\leq (18)/(3)\\\\x&\leq 6\end{aligned}`

So, the solution to the right inequality is x ≤ 6.

Now, we need to find the values of x that satisfy both inequalities simultaneously, which means we are looking for the intersection of the solutions.

Since x must be greater than or equal to 4 (x ≥ 4) and less than or equal to 6 (x ≤ 6), the common solution is 4 ≤ x ≤ 6.

So, the solution to the compound inequality 2 + x ≤ 3x − 6 ≤ 12 is:

`\Large\boxed{\boxed{4 \leq x \leq 6}}`