To solve the compound inequality [tex]\( -2x + 8 < 14 \)[/tex] and [tex]\( -3x - 9 \geq -12 \)[/tex], let’s break it down step-by-step for each inequality separately.
1. Solve the first inequality: [tex]\( -2x + 8 < 14 \)[/tex]
- First, subtract 8 from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
-2x < 14 - 8
\][/tex]
[tex]\[
-2x < 6
\][/tex]
- Next, divide both sides by [tex]\(-2\)[/tex] to solve for [tex]\( x \)[/tex]. Remember that dividing by a negative number reverses the inequality direction:
[tex]\[
x > \frac{6}{-2}
\][/tex]
[tex]\[
x > -3
\][/tex]
2. Solve the second inequality: [tex]\( -3x - 9 \geq -12 \)[/tex]
- First, add 9 to both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
-3x \geq -12 + 9
\][/tex]
[tex]\[
-3x \geq -3
\][/tex]
- Next, divide both sides by [tex]\(-3\)[/tex] to solve for [tex]\( x \)[/tex]. Again, remember that dividing by a negative number reverses the inequality direction:
[tex]\[
x \leq \frac{-3}{-3}
\][/tex]
[tex]\[
x \leq 1
\][/tex]
3. Combine the results:
- From the first inequality, we have [tex]\( x > -3 \)[/tex].
- From the second inequality, we have [tex]\( x \leq 1 \)[/tex].
Combining these two results, we get the following compound inequality:
[tex]\[
-3 < x \leq 1
\][/tex]
Therefore, the solution to the compound inequality [tex]\( -2x + 8 < 14 \)[/tex] and [tex]\( -3x - 9 \geq -12 \)[/tex] is:
[tex]\[
-3 < x \leq 1
\][/tex]
