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Contessa is solving an absolute value inequality. She writes the compound inequality [tex]\( -7 \leq 8-3q \leq 7 \)[/tex] as her first step. Which inequality is Contessa solving?

A. [tex]\( |8-3q| \geq 7 \)[/tex]
B. [tex]\( |8-3q| \geq 0 \)[/tex]
C. [tex]\( |8-3q| \leq 7 \)[/tex]
D. [tex]\( |8-3q| \leq 0 \)[/tex]

Sagot :

GinnyAnswer
To determine which absolute value inequality Contessa is solving, we start with the compound inequality she wrote:

[tex]\[ -7 \leq 8 - 3q \leq 7 \][/tex]

### Step-by-Step Process

1. Recognize the Form of an Absolute Value Inequality:
An absolute value inequality [tex]\(|A| \leq B\)[/tex] can be rewritten as:
[tex]\[ -B \leq A \leq B \][/tex]

Similarly, an absolute value inequality [tex]\(|A| \geq B\)[/tex] can be rewritten as:
[tex]\[ A \leq -B \text{ or } A \geq B \][/tex]

2. Compare the Given Inequality:
Looking at the compound inequality Contessa wrote:
[tex]\[ -7 \leq 8 - 3q \leq 7 \][/tex]

We notice that it matches the standard form of [tex]\(-B \leq A \leq B\)[/tex], where [tex]\(A = 8 - 3q\)[/tex] and [tex]\(B = 7\)[/tex].

3. Determine the Equivalent Absolute Value Inequality:
Given the structured form above, we rewrite the compound inequality as:
[tex]\[ |8 - 3q| \leq 7 \][/tex]

Therefore, the absolute value inequality that Contessa is solving is:

[tex]\(\boxed{|8 - 3q| \leq 7}\)[/tex]

This aligns with the third option provided:

[tex]\[ |8 - 3q| \leq 7 \][/tex]
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