The inequality in the image is:

$|2x| + |3 - 2x| \leq 3$

This inequality involves absolute values, so we need to handle it by considering different cases based on the points where the expressions inside the absolute values change sign.

Step-by-Step Solution:

Case 1: $x \geq \frac{3}{2}$

In this case:

• $|2x| = 2x$ since $2x$ is positive for $x \geq \frac{3}{2}$.
• $|3 - 2x| = 2x - 3$ because $3 - 2x$ becomes negative when $x \geq \frac{3}{2}$.

So, the inequality becomes: $2x + (2x - 3) \leq 3$ Simplifying: $4x - 3 \leq 3$ $4x \leq 6$ $x \leq \frac{3}{2}$ But this case is for $x \geq \frac{3}{2}$, so the only solution for this case is $x = \frac{3}{2}$.

Case 2: $0 \leq x < \frac{3}{2}$

In this case:

• $|2x| = 2x$ since $2x$ is positive for $x \geq 0$.
• $|3 - 2x| = 3 - 2x$ because $3 - 2x$ is positive when $x \leq \frac{3}{2}$.

The inequality becomes: $2x + (3 - 2x) \leq 3$ Simplifying: $2x + 3 - 2x \leq 3$ $3 \leq 3$ This is always true, so $x \in [0, \frac{3}{2}]$ is a solution for this case.

Case 3: $x < 0$

In this case:

• $|2x| = -2x$ because $2x$ is negative for $x < 0$.
• $|3 - 2x| = 3 - 2x$ because $3 - 2x$ is still positive for $x < 0$.

The inequality becomes: $-2x + (3 - 2x) \leq 3$ Simplifying: $-2x + 3 - 2x \leq 3$ $-4x + 3 \leq 3$ $-4x \leq 0$ $x \geq 0$ But this case is for $x < 0$, so there is no solution in this case.

Final Solution:

The solution to the inequality is $x \in [0, \frac{3}{2}]$.

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Would you like further details on any step, or do you have any questions?

Here are some related questions you might explore:

1. How do you solve inequalities involving absolute values?
2. What are the conditions under which absolute value expressions change sign?
3. How do you handle systems of inequalities?
4. What is the graphical representation of this inequality?
5. Can this problem be solved using a different method, like algebraic manipulation?

Tip: When solving absolute value inequalities, always consider the points where each absolute value expression changes sign, and break the problem into cases based on those points.