To solve the inequality \( 2|2x - 3| < |x + 10| \), we need to break it into different cases based on the expressions inside the absolute value signs.

### Step 1: Set up the cases
The absolute value function creates different scenarios depending on whether the expression inside is positive or negative. For the expression \( |2x - 3| \) and \( |x + 10| \), we will examine the following cases:

#### Case 1: \( 2x - 3 \geq 0 \) and \( x + 10 \geq 0 \)
- Here, \( |2x - 3| = 2x - 3 \) and \( |x + 10| = x + 10 \).

The inequality becomes:
\[
2(2x - 3) < x + 10
\]
Simplifying:
\[
4x - 6 < x + 10
\]
Subtract \( x \) from both sides:
\[
3x - 6 < 10
\]
Add 6 to both sides:
\[
3x < 16
\]
Divide by 3:
\[
x < \frac{16}{3}
\]
Since this case assumes \( 2x - 3 \geq 0 \), this implies \( x \geq \frac{3}{2} \).

Therefore, the solution for this case is \( \frac{3}{2} \leq x < \frac{16}{3} \).

#### Case 2: \( 2x - 3 \geq 0 \) and \( x + 10 < 0 \)
- Here, \( |2x - 3| = 2x - 3 \) and \( |x + 10| = -(x + 10) = -x - 10 \).

The inequality becomes:
\[
2(2x - 3) < -x - 10
\]
Simplifying:
\[
4x - 6 < -x - 10
\]
Add \( x \) to both sides:
\[
5x - 6 < -10
\]
Add 6 to both sides:
\[
5x < -4
\]
Divide by 5:
\[
x < -\frac{4}{5}
\]
However, this case assumes \( x + 10 < 0 \), meaning \( x < -10 \). The solution \( x < -\frac{4}{5} \) does not satisfy \( x < -10 \), so **there is no solution** in this case.

#### Case 3: \( 2x - 3 < 0 \) and \( x + 10 \geq 0 \)
- Here, \( |2x - 3| = -(2x - 3) = -2x + 3 \) and \( |x + 10| = x + 10 \).

The inequality becomes:
\[
2(-2x + 3) < x + 10
\]
Simplifying:
\[
-4x + 6 < x + 10
\]
Add \( 4x \) to both sides:
\[
6 < 5x + 10
\]
Subtract 10 from both sides:
\[
-4 < 5x
\]
Divide by 5:
\[
x > -\frac{4}{5}
\]
This case assumes \( 2x - 3 < 0 \), which implies \( x < \frac{3}{2} \).

Therefore, the solution for this case is \( -\frac{4}{5} < x < \frac{3}{2} \).

#### Case 4: \( 2x - 3 < 0 \) and \( x + 10 < 0 \)
- Here, \( |2x - 3| = -(2x - 3) = -2x + 3 \) and \( |x + 10| = -(x + 10) = -x - 10 \).

The inequality becomes:
\[
2(-2x + 3) < -x - 10
\]
Simplifying:
\[
-4x + 6 < -x - 10
\]
Add \( 4x \) to both sides:
\[
6 < 3x - 10
\]
Add 10 to both sides:
\[
16 < 3x
\]
Divide by 3:
\[
x > \frac{16}{

Explore the step-by-step solution to the inequality 2|2x - 3| < |x + 10| using case analysis of absolute value functions. Learn the methods for solving inequalities with absolute values, perfect for students in Grades 10-12.

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