Question

solve the three - part linear inequality express the solution in interval notation
$2\leq - 6-(a + 4)\leq5$

select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. the solution set in interval notation is
(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression )
b. the solution set is the empty set.

Explanation:

Step1: Simplify the middle expression

First, simplify \(-6 - (a + 4)\). Using the distributive property, we get \(-6 - a - 4 = -a - 10\). So the inequality becomes \(2 \leq -a - 10 \leq 5\).

Step2: Add 10 to all parts

To isolate the term with \(a\), we add 10 to all three parts of the inequality.
\[
2 + 10 \leq -a - 10 + 10 \leq 5 + 10
\]
Simplifying each part: \(12 \leq -a \leq 15\).

Step3: Multiply by -1 (and reverse inequalities)

When we multiply an inequality by a negative number, the direction of the inequality signs reverses. Multiply each part by \(-1\):
\[
12 \times (-1) \geq -a \times (-1) \geq 15 \times (-1)
\]
Simplifying gives \(-12 \geq a \geq -15\), which can be rewritten as \(-15 \leq a \leq -12\).

Answer:

A. The solution set in interval notation is \([-15, -12]\)