To solve the inequality [tex]\( |2x - 6| < 10 \)[/tex], we need to approach it step by step, breaking it down into two separate inequalities. This comes from the definition of absolute value, which states that [tex]\( |A| < B \)[/tex] can be written as [tex]\(-B < A < B\)[/tex]. In this case, [tex]\( A = 2x - 6 \)[/tex] and [tex]\( B = 10 \)[/tex].
1. Write the compound inequality:
[tex]\[
-10 < 2x - 6 < 10
\][/tex]
2. We will solve each part of the compound inequality separately.
a. Start with the left part of the inequality:
[tex]\[
-10 < 2x - 6
\][/tex]
- Add 6 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-10 + 6 < 2x
\][/tex]
[tex]\[
-4 < 2x
\][/tex]
- Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
-2 < x
\][/tex]
So, the solution to the left part is:
[tex]\[
x > -2
\][/tex]
b. Next, solve the right part of the inequality:
[tex]\[
2x - 6 < 10
\][/tex]
- Add 6 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x - 6 + 6 < 10 + 6
\][/tex]
[tex]\[
2x < 16
\][/tex]
- Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < 8
\][/tex]
3. Combine the two parts of the compound inequality:
[tex]\[
-2 < x < 8
\][/tex]
Therefore, the solution to the inequality [tex]\( |2x - 6| < 10 \)[/tex] is [tex]\( \{ x \mid -2 < x < 8 \} \)[/tex].
Given the choices:
- [tex]\( \{ x \mid -2 < x < 8 \} \)[/tex]
- [tex]\( \{ x \mid 2 < x < 8 \} \)[/tex]
- [tex]\( \{ x \mid x < -2 \)[/tex] or [tex]\( x > 8 \} \)[/tex]
The correct answer is:
[tex]\( \{ x \mid -2 < x < 8 \} \)[/tex]
