Let's break down and solve each part of the problem step-by-step.

1. Solving the Equation $|2x - 1| = 2$

For an absolute value equation $|A| = B$, we split it into two cases:

1. $2x - 1 = 2$
2. $2x - 1 = -2$

Case 1: $2x - 1 = 2$

1. Add 1 to both sides: $2x = 3$
2. Divide by 2: $x = \frac{3}{2}$

Case 2: $2x - 1 = -2$

1. Add 1 to both sides: $2x = -1$
2. Divide by 2: $x = -\frac{1}{2}$

So, the solutions to $|2x - 1| = 2$ are: $x = \frac{3}{2} \text{ and } x = -\frac{1}{2}$

2. Solving the Inequality $|2x - 1| < 2$

For an absolute value inequality $|A| < B$, we can write it as a compound inequality: $-2 < 2x - 1 < 2$

Now, solve this compound inequality step-by-step.

1. Left Inequality: $-2 < 2x - 1$

   • Add 1 to both sides: $-1 < 2x$
   • Divide by 2: $-\frac{1}{2} < x$

2. Right Inequality: $2x - 1 < 2$

   • Add 1 to both sides: $2x < 3$
   • Divide by 2: $x < \frac{3}{2}$

So, the solution to $|2x - 1| < 2$ is: $-\frac{1}{2} < x < \frac{3}{2}$

Answers

• For the equation $|2x - 1| = 2$: The solutions are $x = \frac{3}{2}$ and $x = -\frac{1}{2}$.
• For the inequality $|2x - 1| < 2$: The solution set is $-\frac{1}{2} < x < \frac{3}{2}$, which matches option B.

Would you like further details on these steps or have additional questions?

Related Questions

1. How would the solution change if the inequality was $|2x - 1| > 2$?
2. How do we solve absolute value equations and inequalities with different constants, like $|3x - 2| = 5$?
3. Can we represent $|2x - 1| = 2$ graphically and interpret the solution visually?
4. What if $|2x - 1|$ had been less than or equal to 2 instead of strictly less?
5. How do we verify solutions for absolute value equations?

Tip

When solving absolute value inequalities, remember that $|A| < B$ represents values of $A$ between $-B$ and $B$, while $|A| > B$ represents values of $A$ outside this range.