Question
15. Complete the solution of \( |2 x-8|<10 \)
Ask by Pollard Parsons. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( -1 < x < 9 \).
Solution
1. Start with the inequality:
\[
|2x - 8| < 10
\]
2. Rewrite the inequality without the absolute value by expressing it as a double inequality:
\[
-10 < 2x - 8 < 10
\]
3. Add \(8\) to all parts of the inequality:
\[
-10 + 8 < 2x - 8 + 8 < 10 + 8
\]
which simplifies to:
\[
-2 < 2x < 18
\]
4. Divide every part by \(2\) to solve for \(x\):
\[
\frac{-2}{2} < \frac{2x}{2} < \frac{18}{2}
\]
which simplifies to:
\[
-1 < x < 9
\]
The solution of the inequality is:
\[
x \in (-1, 9)
\]
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Simplify this solution Bonus Knowledge
To solve the inequality \( |2x - 8| < 10 \), we first split it into two separate inequalities based on the definition of absolute value: 1. \( 2x - 8 < 10 \) 2. \( 2x - 8 > -10 \) For the first inequality \( 2x - 8 < 10 \): - Add 8 to both sides: \( 2x < 18 \) - Divide by 2: \( x < 9 \) For the second inequality \( 2x - 8 > -10 \): - Add 8 to both sides: \( 2x > -2 \) - Divide by 2: \( x > -1 \) Combining these results, we get the solution \( -1 < x < 9 \), or in interval notation, \( (-1, 9) \). Remember, the absolute value creates a "sandwich" effect, so you’re essentially looking for values of \( x \) that fall neatly within those bounds!
