When do you flip the inequality sign?
Updated
The short answer
Flip the inequality sign whenever you multiply or divide both sides by a negative number, so −3x < 12 becomes x > −4. Adding, subtracting, and multiplying or dividing by a positive number all leave the sign as it is.
What it looks like
Here is −3x + 5 < 17 solved two ways.
| Step | Wrong | Right |
|---|---|---|
| Subtract 5 from both sides | −3x < 12 | −3x < 12 |
| Divide both sides by −3 | x < −4 | x > −4 |
You can test the wrong answer with a number it allows. x = −5 is less than −4, but −3(−5) + 5 = 20, and 20 < 17 is false. A number from the right answer works: with x = 0, −3(0) + 5 = 5, and 5 < 17 is true. The first place the work goes wrong is the < in the second line.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
Solving an inequality feels exactly like solving an equation, and equations never make you think about direction. Most steps really are the same. The one difference is multiplying or dividing by a negative number. Paul Dawkins's algebra notes call it the fact that is often the most misused or forgotten in solving inequalities.
Here is why the sign has to turn around. Start with a true statement, −3 < 5. Multiply both sides by −2 and you get 6 and −10, and 6 is bigger than −10. Multiplying by a negative number reverses the order of any two numbers, so the side that was smaller becomes the side that's bigger.
How to catch it yourself
Each time you multiply or divide both sides, look at the number you used. If it's negative, the sign turns around in that same line. Then point at the sign in your new line and ask yourself:
What did dividing by −3 do to the direction of the sign?
To check your answer, pick one number inside it and one outside it, and try both in the original inequality. Only the first should work.
If you'd rather skip the flip, move the x term to the side where it's positive. Adding 3x and subtracting 12 on both sides turns −3x < 12 into −12 < 3x, and dividing by 3 gives −4 < x. That's the same answer as x > −4.
Try these
Solve each one.
- −2x > 10
- 3x < −12
- 4 − x ≤ 9
Answers
- x < −5
- x < −4, with no flip, because you divide by a positive 3
- x ≥ −5, because −x ≤ 5, and dividing by −1 flips the sign
How alai helps
In Homework help, when you pause after writing a step, Ally reads it and compares it with the line before. Ally asks about the first line that doesn't follow and leaves the fix to you. Ally also remembers slips from past sessions, so a later session can start from there.