Solve the inequality \$| (2x + 3)/(x - 4) ge1 | \$.

Rewrite the absolute value inequality as two separate inequalities, first we take the positive value

\$ (2x + 3)/(x - 4) ge 1 \$ (or) \$ (2x + 3)/(x - 4) le - 1 \$

\$ (2x + 3)/(x - 4) ge 1 \$

Multiply both sides by x - 4 (assuming \$ x ne 4\$, since division by zero is undefined), then simplify

\$ (2x + 3)/(\cancel(x - 4)) times \cancel(x - 4) ge 1 times (x - 4) \$

\$ 2x + 3 ge x - 4 \$

Move the common terms to the same side then simplify

\$ 2x - x ge - 4 - 3 \$

\$ x ge - 7 \$

Second, we take the negative value and then simplify

\$ (2x + 3)/(x - 4) le - 1 \$

\$ (2x + 3)/\cancel(x - 4) times \cancel(x - 4) le -1 times (x - 4) \$

\$ 2x + 3 le - (x - 4) \$

After simplification

\$ 2x + 3 le - x + 4 \$

\$ 2x + x le 4 - 3 \$

\$ 3x le 1 \$

\$ x le 1/3 \$

So, the solution to the inequality \$| (2x + 3)/(x - 4) ge1 | \$ is \$-7 le x le 1/3\$.

This reads x is less than or equal to -7, which is less than or equal to \$1/3\$. This doesn’t represent the solution because -7 is not less than or equal to \$1/3\$

\$x le -7 le 1/3\$

This reads x is greater than or equal to -7, greater than or equal to 1/3. This flips both inequalities and doesn’t represent the solution

\$x ge -7 ge 1/3\$

This reverses the order of the bounds and flips the first inequality. It doesn’t represent the solution

\$7 ge x le -1/3\$

Option D accurately captures the solution where x can be any value less than or equal to -7, or any value greater than or equal to \$1/3\$

\$-7 le x le 1/3\$

Option

D

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