Part 19 of 20 · InequalitiesSee the whole course
How to solve a linear inequality (and when to flip the sign)
Solve an inequality exactly like an equation, with one new rule: when you multiply or divide both sides by a negative number, flip the inequality sign. So −3x < 12 becomes x > −4.
An inequality has a range of answers
An equation like 2x + 1 = 5 has one answer. An inequality like 2x + 1 > 5 uses a comparison sign (<, >, ≤, ≥), and its answer is a whole range of numbers: x > 2 means every number bigger than 2.
Solve it like an equation
Do the same thing to both sides to get x alone, exactly as in solving a linear equation.
2x + 1 > 5
Subtract 1: 2x > 4
Divide by 2: x > 2
Check: x = 3 gives 2(3) + 1 = 7, and 7 > 5 ✓. x = 0 gives 1, and 1 > 5 is false ✓.
On a graph, 2x + 1 > 5 means “the line y = 2x + 1 is above the line y = 5”. That happens to the right of where they cross, at x = 2.
The one new rule: flip for a negative
Adding, subtracting, and multiplying or dividing by a positive number never change the sign. But multiplying or dividing by a negative number reverses the order of the numbers. 2 < 5 is true, but −2 < −5 is false: −2 > −5. So the sign must flip.
−3x < 12
Divide both sides by −3 and flip the sign: x > −4
Check: x = 0 gives −3(0) = 0, and 0 < 12 ✓. x = −5 gives −3(−5) = 15, and 15 < 12 is false ✓.
Switch between the two inequalities below. Watch the green zone: when you divide by −3, the zone moves to the other side of x = −4.
- 2x + 1 > 5
- 2x > 4 (subtract 1)
- x > 2 (divide by 2)
- test x = 3: 2(3) + 1 = 7 > 5 ✓
Two more examples
3x − 4 ≤ 8: add 4 to get 3x ≤ 12, then divide by 3: x ≤ 4 (no flip, 3 is positive)
−2x + 5 ≥ 1: subtract 5 to get −2x ≥ −4, then divide by −2 and flip: x ≤ 2
Notice that ≤ flips to ≥, and < flips to >. The “or equal to” part stays.
Watch it explained in under a minute
The four mistakes to avoid
- Forgetting to flip when dividing by a negative.
- Flipping when you only add or subtract. Only multiplying or dividing by a negative flips the sign.
- Flipping the wrong way. < becomes >, and ≤ becomes ≥.
- Skipping the test number. Put one number from your answer back in to see that it works.
Try it yourself
Tap a question to see the answer.
1. 3x − 2 < 10
Add 2: 3x < 12. Divide by 3: x < 4. No flip.
2. −2x > 6
Divide by −2 and flip: x < −3. Check x = −4: −2(−4) = 8, and 8 > 6 ✓.
3. 5 − x ≥ 2
Subtract 5: −x ≥ −3. Divide by −1 and flip: x ≤ 3. Check x = 0: 5 ≥ 2 ✓, and x = 4: 1 ≥ 2 is false ✓.
Where this fits in
You can now find which numbers an inequality allows. Next, the picture of the answer: how to graph an inequality.