To solve the given inequality [tex]\(|2x + 10| > 6\)[/tex], let's break it down into two separate cases based on the properties of absolute values.
1. First Case: [tex]\(2x + 10 > 6\)[/tex]
- Start with the inequality [tex]\(2x + 10 > 6\)[/tex].
- To isolate [tex]\(x\)[/tex], first subtract 10 from both sides:
[tex]\[
2x + 10 - 10 > 6 - 10
\][/tex]
[tex]\[
2x > -4
\][/tex]
- Next, divide both sides by 2:
[tex]\[
x > -2
\][/tex]
2. Second Case: [tex]\(2x + 10 < -6\)[/tex]
- Start with the inequality [tex]\(2x + 10 < -6\)[/tex].
- To isolate [tex]\(x\)[/tex], first subtract 10 from both sides:
[tex]\[
2x + 10 - 10 < -6 - 10
\][/tex]
[tex]\[
2x < -16
\][/tex]
- Next, divide both sides by 2:
[tex]\[
x < -8
\][/tex]
Combining these two cases, we have the solution to the inequality [tex]\(|2x + 10| > 6\)[/tex]:
[tex]\[
x < -8 \quad \text{or} \quad x > -2
\][/tex]
Thus, the solution is [tex]\(x < -8\)[/tex] or [tex]\(x > -2\)[/tex].
