Question

Eduardo solved the following inequality, and his work is shown below:

−5(x + 4) + 21 ≥ −3 + 4(x − 8)
−5x − 20 + 21 ≥ −3 + 4x − 32
−5x + 1 ≥ 4x − 35
−9x ≥ −36
x ≥ 4

What mistake did Eduardo make in solving the inequality?

Answer 1

Hint: It's the last step.

Answer 2

−5(x + 4) + 21 ≥ −3 + 4(x − 8)

Use the distributive property on both sides to get:

−5x − 20 + 21 ≥ −3 + 4x − 32 <----- same as Eduardo's

Combine like terms on each side.

−5x + 1 ≥ 4x − 35 <----- same as Eduardo's

Subtract 1 from both sides.
Subtract 4x from both sides.

−9x ≥ −36 <---- same as Eduardo's

Up to here Eduardo is correct.

Answer 3

How do you divide both sides of an inequality by a NEGATIVE number?

Answer 4

the sign flips?

Answer 5

Look at the following.

Let's say that this is a true statement.

\(\Large x \gt 5\)

Let's think of a number that makes the above inequality true.
For example, let x = 7.
x = 7 makes the inequality true since \(\Large 7 \gt 5\)

Now divide both sides by -1:

\(\Large -x \gt -5\)

If this new inequality is equivalent to teh original one, then x = 7 still has to work.
Let's try it.

We plug in 7 for x. Since the left side has -x, we get -7 on the left side.

\(\Large -7 \gt -5\)

This is a false statement, so you cannot just divide both sides of an inequality by -1.

What would make the inequality true?

\(\Large -7 \lt -5\)

is a true inequality, but to obtain it, we had to switch the greater than sign for a less than sign.

Answer 6

You are correct.

Answer 7

The above shows that if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Answer 8

−9x ≥ −36

Divide both sides of the inequality by -9 and flip the sign.

\(\large x \le 4\) <----- correct solution, not like Eduardo's