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

When \$3x - 2 ge 0\$ and \$ 2x + 1 ge 0 \$ (both absolute values are positive).

In this case, the inequality takes a positive value and then simplifies the inequality

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

5x - 1 < 5

\$5x - \cancel1 + \cancel1 < 5 + 1\$

After simplification

\$ 5x < 6 \$

\$ x < 6/5 \$

When \$3x - 2 < 0 and 2x + 1 ge 0\$ (the first absolute value is negative, the second is positive).

In this case, the inequality takes a positive and negative value and then simplifies the inequality

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

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

-x + 3 < 5

Subtracting 3 from both sides, then we get

\$- x + \cancel3 - \cancel3 < 5 - 3\$

-x < 2

Multiplying both sides by - 1 (remember to reverse the inequality when multiplying or dividing by a negative number), we have

x > - 2

When 3x - 2 < 0 and 2x + 1 < 0 (both absolute values are negative)

In this case, the inequality takes a negative value and then simplifies the inequality

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

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

-5x + 1 < 5

Subtracting 1 from both sides, then simplify

\$ - 5x + \cancel1 - \cancel1 < 5 - 1 \$

\$ - 5x < 4 \$

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

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

Option A is incorrect because it incorrectly states that x > 2, which is not part of the solution set

\$ x < 6/5 \$, \$ x > 2\$ and \$ x < 4/5 \$

Option B is wrong as it claims \$x > 6/5\$ and \$x < −4/5\$, contradicting the obtained solutions

\$ x > 6/5 \$, \$ x > - 2\$ and \$ x < - 4/5 \$

Wrong due to including x > 2, which isn’t a valid solution in analyzed cases

\$ x < 6/5 \$, \$ x > 2\$ and \$ x < - 4/5 \$

Option D is correct because it correctly identifies the ranges for x based on these conditions

\$ x < 6/5 \$, \$ x > - 2\$ and \$ x < - 4/5 \$

Option

D

Can you summarize what you’ve understood in the above steps?