To solve this problem, we’ll break down both the equation and inequality provided:

1. Solve the Equation: $|2x - 3| = 6$

The absolute value equation $|2x - 3| = 6$ can be solved by considering the two possible cases:

Case 1: $2x - 3 = 6$

$2x = 6 + 3 = 9$ $x = \frac{9}{2} = 4.5$

Case 2: $2x - 3 = -6$

$2x = -6 + 3 = -3$ $x = \frac{-3}{2} = -1.5$

So, the solutions to the equation $|2x - 3| = 6$ are: $x = 4.5 \text{ and } x = -1.5$

2. Solve the Inequality: $|2x - 3| < 6$

To solve $|2x - 3| < 6$, we need to set up a compound inequality without the absolute value: $-6 < 2x - 3 < 6$

We can split this into two inequalities and solve each part:

First Inequality: $-6 < 2x - 3$

$-6 + 3 < 2x$ $-3 < 2x$ $x > -\frac{3}{2} = -1.5$

Second Inequality: $2x - 3 < 6$

$2x < 6 + 3$ $2x < 9$ $x < \frac{9}{2} = 4.5$

Combining these two results, we get: $-1.5 < x < 4.5$

Answer Summary

• Solutions to the equation $|2x - 3| = 6$: $x = -1.5, 4.5$.
• Solution to the inequality $|2x - 3| < 6$: $-1.5 < x < 4.5$.

Multiple Choice Selection

Based on the solution for the inequality:

• The correct answer for the inequality $|2x - 3| < 6$ is Option B: $-1.5 < x < 4.5$.

Would you like further details or have any questions?

Related Questions

1. How do you graph absolute value equations and inequalities?
2. What is the general approach to solving absolute value equations?
3. How can we verify the solutions to an absolute value inequality graphically?
4. What are some applications of absolute value equations and inequalities in real-life situations?
5. How do the solutions differ when the inequality is $|2x - 3| \leq 6$?

Tip

When solving absolute value inequalities, remember to set up a compound inequality when the inequality is less than or less than or equal to.