What is the complete solution set, expressed in interval notation, for the inequality \lef… — ACT

What is the complete solution set, expressed in interval notation, for the inequality $\left| \frac{2-3t}{4} \right| > 2$ ?

A
$(-\infty, -2)$
B
$\left(-2, \frac{10}{3}\right)$
C
$(-\infty, -8) \cup (8, \infty)$
$(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)$ Cevap
E
$\left(-\infty, -\frac{10}{3}\right) \cup (2, \infty)$

The complete solution set is

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

The correct answer shows

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

because solving the absolute value inequality

|2-3t| > 8

requires splitting it into two inequalities:

2-3t > 8

and

2-3t < -8

. Solving the first yields

t < -2

after reversing the inequality sign when dividing by

-3

. Solving the second yields

t > \frac{10}{3}

after also reversing the inequality sign. The union of these two intervals is

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

Multiply both sides of the inequality by

4

to isolate the absolute value term.

|2-3t| > 8

Eliminating the denominator simplifies the absolute value expression.

Split the absolute value inequality into two separate linear inequalities representing the positive and negative cases.

2-3t > 8

2-3t < -8

An absolute value greater than a positive number

c

is equivalent to the expression being greater than

c

or less than

-c

Solve the first inequality: subtract

2

from both sides, then divide by

-3

and reverse the inequality sign.

-3t > 6 \implies t < -2

Dividing an inequality by a negative number requires flipping the inequality sign.

Solve the second inequality: subtract

2

from both sides, then divide by

-3

and reverse the inequality sign.

-3t < -10 \implies t > \frac{10}{3}

Dividing an inequality by a negative number requires flipping the inequality sign.

Combine the two solutions using interval notation.

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

The union of the two intervals represents the complete set of values that satisfy either inequality.

Solving absolute value inequalities of the form

|ax+b| > c

by splitting them into two cases and reversing the inequality sign when dividing by a negative number.

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