Question

solve the following absolute value inequality.

\\|2x - 5| > 9\\

\\(x > ?\\) or \\(x < \\)

Explanation:

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<reasoning>

Set up the compound inequalities

An absolute value inequality of the form \(|u| > a\) (where \(a > 0\)) splits into a compound inequality using "or":
\[2x - 5 > 9 \quad \text{or} \quad 2x - 5 < -9\]

Solve the first inequality

Using the Solving Inequalities knowledge point:
\[

$$\begin{aligned} 2x - 5 &> 9 \\ 2x &> 14 \\ x &> 7 \end{aligned}$$

\]

Solve the second inequality

Using the Solving Inequalities knowledge point:
\[

$$\begin{aligned} 2x - 5 &< -9 \\ 2x &< -4 \\ x &< -2 \end{aligned}$$

\]

Combine the solution sets

Using the Compound Inequalities knowledge point, we combine the two individual solutions:
\[x > 7 \quad \text{or} \quad x < -2\]
</reasoning>

<answer>
Solve the following absolute value inequality.
\[|2x - 5| > 9\]
\(x >\) <blank>7</blank> or \(x <\) <blank>-2</blank>
</answer>

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"type": "bentinequality",
"params": [{ "x": 7, "operator": ">", "labelText": "x > 7", "strokeColor": "#8C55F2" }]
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"question_type": "Fill-in-the-blank",
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Answer:

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</pre_analysis>

<reasoning>

Set up the compound inequalities

An absolute value inequality of the form \(|u| > a\) (where \(a > 0\)) splits into a compound inequality using "or":
\[2x - 5 > 9 \quad \text{or} \quad 2x - 5 < -9\]

Solve the first inequality

Using the Solving Inequalities knowledge point:
\[

$$\begin{aligned} 2x - 5 &> 9 \\ 2x &> 14 \\ x &> 7 \end{aligned}$$

\]

Solve the second inequality

Using the Solving Inequalities knowledge point:
\[

$$\begin{aligned} 2x - 5 &< -9 \\ 2x &< -4 \\ x &< -2 \end{aligned}$$

\]

Combine the solution sets

Using the Compound Inequalities knowledge point, we combine the two individual solutions:
\[x > 7 \quad \text{or} \quad x < -2\]
</reasoning>

<answer>
Solve the following absolute value inequality.
\[|2x - 5| > 9\]
\(x >\) <blank>7</blank> or \(x <\) <blank>-2</blank>
</answer>

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