Let's solve the equation step-by-step and verify Jesse's work to determine the correct answer.
The given equation is:
[tex]\[ 4 + |2x + 5| = 8 \][/tex]
Step 1: Isolate the absolute value term.
[tex]\[
|2x + 5| = 8 - 4
\][/tex]
[tex]\[
|2x + 5| = 4
\][/tex]
Step 2: Set up two separate equations based on the definition of absolute value.
The equation [tex]\( |A| = B \)[/tex] implies [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Therefore:
[tex]\[
2x + 5 = 4 \quad \text{or} \quad 2x + 5 = -4
\][/tex]
Step 3: Solve each equation for [tex]\( x \)[/tex].
For [tex]\( 2x + 5 = 4 \)[/tex]:
[tex]\[
2x + 5 = 4
\][/tex]
[tex]\[
2x = 4 - 5
\][/tex]
[tex]\[
2x = -1
\][/tex]
[tex]\[
x = -\frac{1}{2}
\][/tex]
For [tex]\( 2x + 5 = -4 \)[/tex]:
[tex]\[
2x + 5 = -4
\][/tex]
[tex]\[
2x = -4 - 5
\][/tex]
[tex]\[
2x = -9
\][/tex]
[tex]\[
x = -\frac{9}{2}
\][/tex]
So the solutions are:
[tex]\[
x = -\frac{1}{2} \quad \text{or} \quad x = -\frac{9}{2}
\][/tex]
Step 4: Verify the solutions are correct.
Substitute [tex]\( x = -\frac{1}{2} \)[/tex] into the original equation to check:
[tex]\[
4 + |2\left(-\frac{1}{2}\right) + 5| = 4 + |-1 + 5| = 4 + |4| = 4 + 4 = 8
\][/tex]
This solution is correct.
Substitute [tex]\( x = -\frac{9}{2} \)[/tex] into the original equation to check:
[tex]\[
4 + |2\left(-\frac{9}{2}\right) + 5| = 4 + |-9 + 5| = 4 + |-4| = 4 + 4 = 8
\][/tex]
This solution is also correct.
Therefore, Jesse's solutions [tex]\( x = -\frac{1}{2} \)[/tex] and [tex]\( x = -\frac{9}{2} \)[/tex] are correct.
The correct answer is therefore:
[tex]\[ \text{D. Jesse correctly solved the equation.} \][/tex]
