https://file.aiquickdraw.com/mathbot/file/1726715363755ml3klsbf.jpg

The inequality in the image is:

\[
|4x - 2| \leq 10
\]

This is an absolute value inequality, which can be solved in two cases:

1. \( 4x - 2 \leq 10 \)
2. \( -(4x - 2) \leq 10 \) (or equivalently \( 4x - 2 \geq -10 \))

Let's solve each case:

### Case 1: \( 4x - 2 \leq 10 \)
Add 2 to both sides:
\[
4x \leq 12
\]
Now, divide both sides by 4:
\[
x \leq 3
\]

### Case 2: \( 4x - 2 \geq -10 \)
Add 2 to both sides:
\[
4x \geq -8
\]
Now, divide both sides by 4:
\[
x \geq -2
\]

### Combined solution:
The solution to the inequality is:
\[
-2 \leq x \leq 3
\]

This means the values of \(x\) lie between \(-2\) and \(3\), inclusive.

Would you like further details or explanations?

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Here are some related questions to explore further:

1. How do we graph the solution \( -2 \leq x \leq 3 \)?
2. What changes if the inequality were \( |4x - 2| \geq 10 \) instead?
3. How do we handle absolute value inequalities when there is a strict inequality \( |4x - 2| < 10 \)?
4. What is the general method for solving absolute value equations or inequalities?
5. Can we express this inequality solution in interval notation?

**Tip:** When solving absolute value inequalities, always split the problem into two cases (positive and negative).

Learn how to solve the absolute value inequality |4x - 2| ≤ 10 with a step-by-step guide. This algebraic problem demonstrates solving inequalities by splitting them into two cases. Suitable for students in Grades 8-10.

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