Where Did I Go Wrong When Solving Inequalities With Negative Numbers?

[paulb203]

Homework Statement

Solve for x
4>19-3x

Relevant Equations

N/A

My attempt:

4>19-3x

Subtract 19 from both sides:

-15 > -3x

Divide both sides by -3:

5 > x

Switch sides (change sign):

x < 5

! But Maths Genie tells me the answer is x>5

Where have I gone wrong?

 

[Hill]

You did not divide by -3 correctly.

 

[fresh_42]

Homework Statement: Solve for x
4>19-3x
Relevant Equations: N/A

My attempt:

4>19-3x

Subtract 19 from both sides:

-15 > -3x

Divide both sides by -3:

5 > x

This is wrong. If you multiply or divide by a negative number, then you have to reverse the '>' sign. Thus it has to be 5 < x.

Switch sides (change sign):

x < 5

You switched sign AND sides, so you didn't change anything. 5 < 6 and 6 > 5 are the same.

! But Maths Genie tells me the answer is x>5

Where have I gone wrong?

 

[PeroK]

Homework Statement: Solve for x
4>19-3x
Relevant Equations: N/A

My attempt:

4>19-3x

Subtract 19 from both sides:

-15 > -3x

Divide both sides by -3:

5 > x

Switch sides (change sign):

x < 5

! But Maths Genie tells me the answer is x>5

Where have I gone wrong?

One way to find your error yourself is to check each step with numbers that should meet the inequality.

We start with $4 > 19-3x$.
Try $x = 4$. We have $19 -3x = 7$, so the inequality fails.
Try $x = 6$. We have $19 -3x = 1$, so the inequality holds.

Your next inequality was $-15 > -3x$.
Try $x = 4$. We have $-3x = -12$, so the inequality fails.
Try $x = 6$. We have $-3x = -18$, so the inequality holds. You are still all right.

Then you have $5 > x$. By trying $x = 4$ and $x = 6$, we can see that at this step things have gone wrong, and it should be $5 < x$.

 

[paulb203]

You did not divide by -3 correctly.

Thanks, Hill

Are you talking about the issue with the sign?

Is my division correct on its own, putting the issue with the sign to one side for the moment;

-15 / -3 = 5 ?

-3x / -3 = x ?

 

[Hill]

Are you talking about the issue with the sign?

Yes.

 

[paulb203]

“If you multiply or divide by a negative number, then you have to reverse the '>' sign. “

Ah, thanks, I think I remember this now.

Is this correct:

Example:

-15 > -18

both sides by -3

5 > 6 (incorrect, because they are now positive numbers, and 5 is obv. not greater than 6)

 

[paulb203]

Thanks, peroK

 

[PeroK]

“If you multiply or divide by a negative number, then you have to reverse the '>' sign. “

Ah, thanks, I think I remember this now.

Is this correct:

Example:

-15 > -18

both sides by -3

5 > 6 (incorrect, because they are now positive numbers, and 5 is obv. not greater than 6)

Yes, exactly. The acid test is that $x > 0$ is equivalent to $-x < 0$ (multiplying or dividing both sides by $-1$).

You have to be careful algebraically as well. The following only applies when $a > 0$:
$$ab > ac \ \Rightarrow \ b > c \ \ \ (a > 0)$$