To solve the equation [tex]\(0.25 |1.5x - 2.25| = 2.25\)[/tex], let's work through it step-by-step:
1. Isolate the absolute value expression:
First, divide both sides by 0.25 to isolate the absolute value.
[tex]\[
\frac{0.25 |1.5x - 2.25|}{0.25} = \frac{2.25}{0.25}
\][/tex]
Simplifying, we get:
[tex]\[
|1.5x - 2.25| = 9
\][/tex]
2. Break down the absolute value into two separate equations:
The expression [tex]\( |1.5x - 2.25| = 9 \)[/tex] means that [tex]\( 1.5x - 2.25 \)[/tex] can be either 9 or -9.
So we have two cases:
Case 1:
[tex]\[
1.5x - 2.25 = 9
\][/tex]
Case 2:
[tex]\[
1.5x - 2.25 = -9
\][/tex]
3. Solve each case separately:
Case 1:
[tex]\[
1.5x - 2.25 = 9
\][/tex]
Add 2.25 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
1.5x = 9 + 2.25
\][/tex]
[tex]\[
1.5x = 11.25
\][/tex]
Divide both sides by 1.5:
[tex]\[
x = \frac{11.25}{1.5}
\][/tex]
[tex]\[
x = 7.5
\][/tex]
Case 2:
[tex]\[
1.5x - 2.25 = -9
\][/tex]
Add 2.25 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
1.5x = -9 + 2.25
\][/tex]
[tex]\[
1.5x = -6.75
\][/tex]
Divide both sides by 1.5:
[tex]\[
x = \frac{-6.75}{1.5}
\][/tex]
[tex]\[
x = -4.5
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\( 0.25 |1.5x - 2.25| = 2.25 \)[/tex] are [tex]\( x = -4.5 \)[/tex] and [tex]\( x = 7.5 \)[/tex].
Therefore, the correct answer is:
[tex]\[
x = -4.5, 7.5
\][/tex]
