Eduardo made a mistake when dividing by -9. When you divide or multiply both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
Let's examine Eduardo's work step by step to identify the mistake:
1. Start with the inequality:
[tex]\[
-5(x + 4) + 21 \geq -3 + 4x - 32
\][/tex]
2. Distribute the -5 on the left side:
[tex]\[
-5x - 20 + 21 \geq -3 + 4x - 32
\][/tex]
3. Simplify both sides:
[tex]\[
-5x + 1 \geq 4x - 35
\][/tex]
4. Combine like terms on each side:
[tex]\[
-5x + 1 \geq 4x - 35
\][/tex]
5. Subtract 4x from both sides:
[tex]\[
-5x - 4x + 1 \geq -35
\][/tex]
[tex]\[
-9x + 1 \geq -35
\][/tex]
6. Subtract 1 from both sides:
[tex]\[
-9x \geq -36
\][/tex]
7. Divide both sides by -9, and don't forget to reverse the inequality sign:
[tex]\[
x \leq 4
\][/tex]
Eduardo did not change the [tex]$\geq$[/tex] to [tex]$\leq$[/tex] when dividing by -9. The correct inequality should be:
[tex]\[
x \leq 4
\][/tex]
Therefore, the mistake Eduardo made was:
"When dividing by -9, he did not change the [tex]$\geq$[/tex] to [tex]$\leq$[/tex]."
