Question

solve for t.
|-2t - 5| - 9 > -3
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3. use inte
fractions, or improper fractions in simplest form.

Explanation:

Step1: Isolate the absolute value

Add 9 to both sides of the inequality \(|-2t - 5| - 9 > -3\) to isolate the absolute value expression.
\(|-2t - 5| - 9 + 9 > -3 + 9\)
\(|-2t - 5| > 6\)

Step2: Solve the absolute value inequality

The absolute value inequality \(|A| > B\) (where \(B>0\)) is equivalent to \(A < -B\) or \(A > B\). Here, \(A = -2t - 5\) and \(B = 6\), so we have two inequalities:

1. \(-2t - 5 < -6\)
2. \(-2t - 5 > 6\)

Step3: Solve the first inequality \(-2t - 5 < -6\)

Add 5 to both sides:
\(-2t - 5 + 5 < -6 + 5\)
\(-2t < -1\)
Divide both sides by -2 (and reverse the inequality sign because we are dividing by a negative number):
\(t > \frac{-1}{-2}\)
\(t > \frac{1}{2}\)

Step4: Solve the second inequality \(-2t - 5 > 6\)

Add 5 to both sides:
\(-2t - 5 + 5 > 6 + 5\)
\(-2t > 11\)
Divide both sides by -2 (and reverse the inequality sign):
\(t < \frac{11}{-2}\)
\(t < -\frac{11}{2}\)

Answer:

\(t < -\frac{11}{2}\) or \(t > \frac{1}{2}\)