How to Solve Linear Inequalities and When to Flip the Sign
Solving an inequality such as 3x - 7 > 8 uses almost the same moves as solving an equation. There is one extra rule, and it is responsible for most lost points: when you multiply or divide both sides by a negative number, the inequality sign flips.
The Common Core high school algebra standards list this skill directly: "Solve linear equations and inequalities in one variable." The grade 7 standards introduce it earlier through word problems of the form px + q > r or px + q < r.
1. The symbols
| Symbol | Meaning | On a number line |
|---|---|---|
| > | greater than | open circle |
| < | less than | open circle |
| ≥ | greater than or equal to | closed (filled) circle |
| ≤ | less than or equal to | closed (filled) circle |
An open circle means the endpoint is not part of the answer. A closed circle means it is.
2. Example 1: no flip needed
Solve 3x - 7 > 8.
- Add 7 to both sides: 3x > 15
- Divide both sides by 3: x > 5
Dividing by a positive number does not change the direction. The answer is every number greater than 5. On a number line, draw an open circle at 5 and shade to the right.
Unlike an equation, the answer is not one number. It is a whole set of numbers, and that is why checking works a little differently (see below).
3. Example 2: dividing by a negative
Solve -2x + 5 ≤ 11.
- Subtract 5 from both sides: -2x ≤ 6
- Divide both sides by -2 and flip the sign: x ≥ -3
The answer is every number greater than or equal to -3. Draw a closed circle at -3 and shade to the right.
4. Why the sign flips
Start with a true statement: -2 < 3.
Multiply both sides by -1. You get 2 and -3. But 2 < -3 is false. The true statement is 2 > -3.
Multiplying by a negative number reflects every value across zero on the number line, so the number that was on the left ends up on the right. The order reverses, and the sign has to reverse with it.
What does not cause a flip:
- Adding or subtracting any number, positive or negative
- Multiplying or dividing by a positive number
- Having a negative answer, such as x > -4
Only multiplying or dividing both sides by a negative number flips the sign. Swapping the two sides also reverses the symbol, because 4 < x says the same thing as x > 4.
5. Example 3: variables on both sides
Solve 4x + 1 ≥ 7x - 8.
- Subtract 7x from both sides: -3x + 1 ≥ -8
- Subtract 1 from both sides: -3x ≥ -9
- Divide by -3 and flip: x ≤ 3
You can avoid the flip by collecting x on the side where its coefficient stays positive. Subtract 4x instead: 1 ≥ 3x - 8, then 9 ≥ 3x, then 3 ≥ x. That is the same answer, x ≤ 3, written from the other side.
6. How to check an inequality answer
Use three test values:
- The boundary. For x ≤ 3, try x = 3: left side 4(3) + 1 = 13, right side 7(3) - 8 = 13. The two sides are equal, which confirms 3 is the boundary. Because the sign includes "equal to," 3 belongs in the answer.
- A value inside the answer. Try x = 0: 1 ≥ -8. True.
- A value outside the answer. Try x = 4: 17 ≥ 20. False, as it should be.
If the inside value fails and the outside value passes, the sign is pointing the wrong way, which usually means a missed flip.
7. Example 4: a word problem
The grade 7 standards include this problem: "As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions."
Let s be the number of sales. "At least" means greater than or equal to.
- Inequality: 50 + 3s ≥ 100
- Subtract 50: 3s ≥ 50
- Divide by 3: s ≥ 16.67 (rounded)
The math says s ≥ 16.67, but you cannot make two-thirds of a sale. Sixteen sales pay 50 + 48 = $98, which falls short. Seventeen sales pay 50 + 51 = $101. So the answer in context is at least 17 sales. The standard asks you to interpret the solution in the context of the problem, and this rounding step is that interpretation.
8. Example 5: a compound inequality
Solve -4 < 2x + 2 ≤ 10.
Do the same step to all three parts:
- Subtract 2: -6 < 2x ≤ 8
- Divide by 2: -3 < x ≤ 4
On a number line: open circle at -3, closed circle at 4, shading between them.
9. Looking ahead: two variables
In high school, the same ideas extend to inequalities with two variables. The standards describe graphing the solutions to a linear inequality in two variables "as a half-plane (excluding the boundary in the case of a strict inequality)." The boundary is a line instead of a point, drawn dashed for > or < and solid for ≥ or ≤, but the open-versus-closed logic is the same.
10. Key takeaways
- Solve an inequality the same way as an equation, with one extra rule.
- Flip the sign only when you multiply or divide both sides by a negative number.
- Use an open circle for > and <, and a closed circle for ≥ and ≤.
- Check the boundary, one value inside the answer, and one value outside it.
- In word problems, read the answer in context, such as rounding up to whole sales.