The given inequality is $|2x - 3| \leq 12$. This is an absolute value inequality, and it can be split into two separate inequalities to solve for $x$:

Step 1: Rewrite the inequality

The absolute value inequality $|A| \leq B$ implies: $-B \leq A \leq B$ Here, $A = 2x - 3$ and $B = 12$.

So: $-12 \leq 2x - 3 \leq 12$

Step 2: Solve the compound inequality

Now, solve this compound inequality step by step.

1. Add 3 to all parts of the inequality: $-12 + 3 \leq 2x - 3 + 3 \leq 12 + 3$ $-9 \leq 2x \leq 15$

2. Divide through by 2: $\frac{-9}{2} \leq \frac{2x}{2} \leq \frac{15}{2}$ $-\frac{9}{2} \leq x \leq \frac{15}{2}$

Final Solution:

The solution is: $x \in \left[-\frac{9}{2}, \frac{15}{2}\right]$

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Do you want details about the graph of this inequality or have any questions?

5 Related Questions:

1. What happens if the inequality changes to $|2x - 3| < 12$?
2. How do you solve $|2x - 3| \geq 12$?
3. How do absolute value inequalities differ when the direction is $\geq$ or $>$?
4. How can this solution be represented on a number line?
5. What are the real-world applications of absolute value inequalities?

Tip:

Always check if the inequality includes equality ($\leq$ or $\geq$) to include the boundary points in the solution.