1

Problem

Solve the inequality \$ | 3x - 2 | + | 2x + 1 | < 5 \$.

2

Step

THe given inequality is

\$ | 3x - 2 | + | 2x + 1 | < 5 \$

3

Clue

To solve this, we need to consider the critical points where the expressions inside the absolute values change signs. These points are:

\$3x - 2 = 0 => x = 2/3\$

\$2x + 1 = 0 => x = -1/2\$

4

Step

These points divide the number line into three intervals:

5

Step

Let’s analyze each interval:

\$x < -1/2\$

\$3x - 2 < 0 => | 3x - 2 | = -(3x - 2) = -3x + 2\$

\$2x + 1 < 0 => | 2x + 1 | = -(2x + 1) = -2x - 1\$

6

Step

The inequality becomes:

\$(-3x + 2) + (-2x - 1) < 5\$

\$-5x + 1 < 5\$

\$-5x < 4\$

\$x > -4/5\$

7

Step

Since we are considering \$x < -1/2\$, we need to find the intersection of \$x < -1/2 and x > -4/5\$

\$-4/5 = -0.8\$

\$-1/2 = -0.5\$

So, \$-4/5 < x < -1/2\$

8

Step

Next Interval

\$-1/2 ≤ x < 2/3\$

\$3x - 2 < 0 => | 3x - 2 | = -(3x - 2) = -3x + 2\$

\$2x + 1 ≥ 0 => | 2x + 1 | = 2x + 1\$

The inequality becomes:

(-3x + 2) + (2x + 1) < 5

-x + 3 < 5

-x < 2

x > -2

9

Step

Since we are considering \$-1/2 ≤ x < 2/3\$, we need to find the intersection of \$x > -2 and -1/2 ≤ x < 2/3\$.

Therefore, \$-1/2 ≤ x < 2/3\$.

10

Step

Final Interval

\$x ≥ 2/3\$

\$3x - 2 ≥ 0 => | 3x - 2 | = 3x - 2\$

\$2x + 1 > 0 => |2x + 1 | = 2x + 1\$

The inequality becomes:

(3x - 2) + (2x + 1) < 5

5x - 1 < 5

5x < 6

\$x < 6/5\$

11

Step

Since we are considering \$x ≥ 2/3\$, we need to find the intersection of \$x ≥ 2/3 and x < 6/5\$

\$2/3 = 0.666...\$

\$6/5 = 1.2\$

Therefore, \$2/3 ≤ x < 6/5\$

12

Clue

Now we combine the solutions from each interval:

\$-4/5 < x < -1/2\$

\$-1/2 ≤ x < 2/3\$

\$2/3 ≤ x < 6/5\$

Combining these intervals, we get: \$-4/5 < x < 6/5\$.

13

Solution

The solution to the inequality \$| 3x - 2 | + | 2x + 1 | < 5 is -4/5 < x < 6/5\$.

14

Sumup

Please summarize steps

Choices

15

Choice-A

This option is incorrect. It suggests that x is greater than \$-4/5\$ and less than \$-6/5\$. However, \$-4/5\$ is greater than \$-6/5\$, so no number can satisfy both conditions simultaneously. This is an empty set

Wrong \$(-4/5) < x < -6/5\$

16

Choice-B

This option is a possible range, but it is incorrect because it misses the negative portion of the solution

Wrong \$(4/5) < x < 6/5\$

17

Choice-C

This option has the correct upper bound of \$6/5\$, but the lower bound of \$-4/7\$ is incorrect

Wrong \$(-4/7) < x < 6/5\$

18

Choice-D

This option is the correct solution. It accurately represents the range of x values that satisfy the inequality

Correct \$(-4/5) < x < 6/5\$

19

Answer

Option

D

20

Sumup

Please summarize choices