Solve: [tex]$2|x+7|-4 \geq 0$[/tex]
Express the answer in set-builder notation.
A. [tex]$\left\{x \mid 5\ \textless \ x\ \textless \ 6\right\}$[/tex]
B. [tex]$\left\{x \mid x \geq-5\right\}$[/tex]
C. [tex]$\left\{x \mid x \leq-9$ or $x \geq-5\right\}$[/tex]
D. [tex]$\left\{x \mid-9\ \textless \ x\ \textless \ -5\right\}$[/tex]
Answer (1)
Isolate the absolute value: ∣ x + 7∣ g e q 2 .
Split into two cases: x + 7 g e q 2 and x + 7 ≤ − 2 .
Solve each case: xg e q − 5 and x ≤ − 9 .
Express the solution in set-builder notation: { x mi d x ≤ − 9 or xg e q − 5 } .
Explanation
Problem Analysis We are given the inequality 2∣ x + 7∣ − 4 g e q 0 . Our goal is to solve for x and express the solution in set-builder notation.
Isolating the Absolute Value First, we isolate the absolute value term. Add 4 to both sides of the inequality: 2∣ x + 7∣ g e q 4
Then, divide both sides by 2: ∣ x + 7∣ g e q 2
Considering Cases Now, we consider two cases based on the properties of absolute value.
Case 1: Positive Argument Case 1: x + 7 g e q 2 . Subtract 7 from both sides: xg e q 2 − 7
xg e q − 5
Case 2: Negative Argument Case 2: x + 7 ≤ − 2 . Subtract 7 from both sides: x ≤ − 2 − 7
x ≤ − 9
Expressing the Solution in Set-Builder Notation Combining the two cases, we have x ≤ − 9 or xg e q − 5 . In set-builder notation, this is written as: { x mi d x ≤ − 9 or xg e q − 5 }
Final Answer Therefore, the solution to the inequality 2∣ x + 7∣ − 4 g e q 0 in set-builder notation is { x mi d x ≤ − 9 or xg e q − 5 } .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, consider a machine that produces parts with a specified length of 10 cm. Due to manufacturing tolerances, the actual length may vary slightly. If the tolerance is 2 cm, we can express the acceptable range of lengths using an absolute value inequality: ∣ x − 10∣ ≤ 2 , where x is the actual length of the part. Solving this inequality tells us that the acceptable lengths are between 8 cm and 12 cm. Similarly, in finance, absolute value inequalities can be used to model acceptable deviations from a target investment return.
