If |2x−7|≤8, what is the greatest possible value of |x−7|? A −12 B 0 C 12 D 152

Understand Absolute Value Definition: We are given the inequality |2x−7|≤8. To solve this, we need to consider the definition of absolute value, which states that |a| ≤ b implies -b ≤ a ≤ b. We apply this to our inequality.Separate into Two Inequalities: We rewrite the inequality as two separate inequalities: -8 ≤ 2x−7 ≤ 8.Solve Left Inequality: Now we solve for x in both inequalities. Starting with the left inequality: -8 ≤ 2x−7. Add 7 to both sides to isolate the term with x: -8 + 7 ≤ 2x.Solve Right Inequality: This simplifies to -1 ≤ 2x. Now divide both sides by 2 to solve for x: -1/2 ≤ x.Combine Both Inequalities: Now we solve the right inequality: 2x−7 ≤ 8. Add 7 to both sides: 2x ≤ 15.Find Greatest |x-7| Value: Divide both sides by 2 to solve for x: x ≤ 15/2.Calculate |x-7| at Endpoints: Combining both inequalities, we have -1/2 ≤ x ≤ 15/2. This is the range of values x can take.Identify Calculation Mistake: Now we need to find the greatest possible value of |x−7|. To do this, we need to consider the distance of x from 7 on the number line. The greatest distance will occur at the endpoints of the interval for x.We calculate |x−7| at the endpoints of the interval. First, for x = -1/2: |(-1/2)−7| = |−7.5| = 7.5.Now for x = 15/2: |(15/2)−7| = |7.5| = 7.5.Both endpoints give us the same value for |x−7|, which is 7.5. However, this value is not listed in the answer choices. We must have made a mistake in our calculations.

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If $|2x-7| \leq 8$ , what is the greatest possible value of $|x-7|$ ? $\newline$ (A) $-12$ $\newline$ (B) $0$ $\newline$ (C) $12$ $\newline$ (D) $152$

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Math Problems

Grade 8

Evaluate absolute value expressions

Full solution

Q. If

|2x-7| \leq 8

, what is the greatest possible value of

|x-7|

\newline

(A)

-12

\newline

(B)

0

\newline

(C)

12

\newline

(D)

152

Understand Absolute Value Definition: We are given the inequality $|2x-7|\leq 8$ . To solve this, we need to consider the definition of absolute value, which states that $|a| \leq b$ implies $-b \leq a \leq b$ . We apply this to our inequality.
Separate into Two Inequalities: We rewrite the inequality as two separate inequalities: $-8 \leq 2x-7 \leq 8$ .
Solve Left Inequality: Now we solve for $x$ in both inequalities. Starting with the left inequality: $-8 \leq 2x-7$ . $\newline$ Add $7$ to both sides to isolate the term with $x$ : $-8 + 7 \leq 2x$ .
Solve Right Inequality: This simplifies to $-1 \leq 2x$ . Now divide both sides by $2$ to solve for $x$ : $-\frac{1}{2} \leq x$ .
Combine Both Inequalities: Now we solve the right inequality: $2x-7 \leq 8$ . Add $7$ to both sides: $2x \leq 15$ .
Find Greatest $\lvert x-7 \rvert$ Value: Divide both sides by $2$ to solve for $x$ : $x \leq \frac{15}{2}$ .
Calculate |x\(-7| ext{ at Endpoints: Combining both inequalities, we have } -\frac{ $1$ }{ $2$ } \leq x \leq \frac{ $15$ }{ $2$ }. \text{This is the range of values } x \text{ can take.}
Identify Calculation Mistake: Now we need to find the greatest possible value of $|x-7|$ . To do this, we need to consider the distance of $x$ from $7$ on the number line. The greatest distance will occur at the endpoints of the interval for $x$ .
Identify Calculation Mistake: Now we need to find the greatest possible value of $|x-7|$ . To do this, we need to consider the distance of $x$ from $7$ on the number line. The greatest distance will occur at the endpoints of the interval for $x$ .We calculate $|x-7|$ at the endpoints of the interval. First, for $x = -\frac{1}{2}$ : $|(-\frac{1}{2})-7| = |-7.5| = 7.5$ .
Identify Calculation Mistake: Now we need to find the greatest possible value of $|x-7|$ . To do this, we need to consider the distance of $x$ from $7$ on the number line. The greatest distance will occur at the endpoints of the interval for $x$ .We calculate $|x-7|$ at the endpoints of the interval. First, for $x = -\frac{1}{2}$ : $|(-\frac{1}{2})-7| = |-7.5| = 7.5$ .Now for $x = \frac{15}{2}$ : $|(\frac{15}{2})-7| = |7.5| = 7.5$ .
Identify Calculation Mistake: Now we need to find the greatest possible value of $|x-7|$ . To do this, we need to consider the distance of $x$ from $7$ on the number line. The greatest distance will occur at the endpoints of the interval for $x$ .We calculate $|x-7|$ at the endpoints of the interval. First, for $x = -\frac{1}{2}$ : $|(-\frac{1}{2})-7| = |-7.5| = 7.5$ .Now for $x = \frac{15}{2}$ : $|(\frac{15}{2})-7| = |7.5| = 7.5$ .Both endpoints give us the same value for $|x-7|$ , which is $x$ $0$ . However, this value is not listed in the answer choices. We must have made a mistake in our calculations.