Solve: [tex]$|2 x-1| < 11$[/tex]
Express the solution in set-builder notation.
A. [tex]$\left{x \mid 5 < x < 6\right}$[/tex]
B. [tex]$\left{x \mid-5 < x < 6\right}$[/tex]
C. [tex]$\left{x \mid x < 6\right}$[/tex]
D. [tex]$\left{x \mid-6 < x < 6\right}$[/tex]
Answer (2)
The solution to the inequality ∣2 x − 1∣ < 11 is found by rewriting it as a compound inequality, leading to the solution in set-builder notation: { x ∣ − 5 < x < 6 } . Therefore, the correct answer is option B.
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Rewrite the absolute value inequality as a compound inequality: − 11 < 2 x − 1 < 11 .
Add 1 to all parts of the inequality: − 10 < 2 x < 12 .
Divide all parts of the inequality by 2: − 5 < x < 6 .
Express the solution in set-builder notation: { x ∣ − 5 < x < 6 } .
Explanation
Understanding the Problem We are asked to solve the absolute value inequality ∣2 x − 1∣ < 11 and express the solution in set-builder notation. This means we need to find all values of x that satisfy the inequality.
Rewriting the Inequality The absolute value inequality ∣2 x − 1∣ < 11 can be rewritten as a compound inequality: − 11 < 2 x − 1 < 11
Isolating the Term with x To solve for x , we first add 1 to all parts of the inequality: − 11 + 1 < 2 x − 1 + 1 < 11 + 1 which simplifies to − 10 < 2 x < 12
Solving for x Next, we divide all parts of the inequality by 2: 2 − 10 < 2 2 x < 2 12 which simplifies to − 5 < x < 6
Expressing the Solution in Set-Builder Notation The solution to the inequality is all values of x such that − 5 < x < 6 . In set-builder notation, this is written as { x ∣ − 5 < x < 6 }
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, suppose a machine is designed to fill bags with 500 grams of sugar, but the actual amount can vary by up to 5 grams. This situation can be modeled by the inequality ∣ x − 500∣ ≤ 5 , where x is the actual weight of the sugar in the bag. Solving this inequality tells us the range of possible weights: 495 ≤ x ≤ 505 . This helps ensure that the machine operates within acceptable limits.
