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Solving a linear equation is just undoing operations until the variable is alone. Inequalities work exactly the same way, with one extra rule that costs students points constantly: when you multiply or divide by a negative, the inequality sign flips. The same undoing works once a variable sits inside absolute-value bars, except that you have to solve two cases instead of one.
Key takeaways
- Isolate the variable by undoing operations in reverse order.
- Inequalities solve like equations, except one rule.
- Multiply or divide by a negative → flip the inequality sign.
- Whatever you do to one side, do to the other.
The one rule that matters
Equations are mechanical: subtract, divide, done. Inequalities are identical, until you divide (or multiply) both sides by a negative number. The moment you do, the < becomes > (or vice versa). Miss that and your answer points the wrong way. (Unsure of the direction? The built-in Desmos calculator can graph both sides to confirm it.)
Dividing by -2? The inequality sign reverses. This single rule is the most common inequality mistake on the test. A wrong-direction answer is always offered.
Try one
Solve the inequality -2x + 3 > 11.
The method
- 1
Undo addition/subtraction first
Move constant terms to the other side.
- 2
Undo multiplication/division
Divide to isolate the variable, and flip the sign if you divided by a negative.
- 3
Check with a number
Plug a value from your solution back in to confirm the direction.
Translate the wording
Inequality word problems are mostly translation, and the test rewards knowing these cold: two of the four choices usually differ only in the direction or strictness of the sign.
| When the problem says… | Write… |
|---|---|
| at least 12 / a minimum of 12 | x \geq 12 |
| at most 12 / no more than 12 | x \leq 12 |
| more than 12 / exceeds 12 | x > 12 |
| fewer than 12 / under 12 | x < 12 |
When an equation has no solution (or infinitely many)
The SAT also likes equations that collapse. Simplify both sides; if the variable cancels and leaves a true statement (9 = 9), every value works: infinitely many solutions. If it leaves a false one (9 = 5), nothing works: no solution. Questions usually hand you a constant and ask which value triggers a case, so line up the variable terms and see what the constants must do.
In the equation 2(3x + c) = 6x + 9, where c is a constant, for what value of c does the equation have infinitely many solutions?
Drill solving and the flip rule with a tutor that catches a wrong-direction answer.
Practice equations & inequalities →Frequently asked questions
- When do you flip the inequality sign?
- Whenever you multiply or divide both sides of an inequality by a negative number. The sign reverses (for example, > becomes <).
- How do you solve a linear equation on the SAT?
- Isolate the variable by undoing operations in reverse order: handle addition and subtraction first, then multiplication and division, doing the same to both sides.
- What’s the most common inequality mistake?
- Forgetting to flip the inequality sign when dividing or multiplying by a negative, which gives an answer pointing the wrong direction.
- What does it mean when a linear equation has no solution?
- After simplifying, the variable terms cancel and you’re left with a false statement like 9 = 5; no value of x can fix it. If you’re left with a true statement instead, every value of x works: infinitely many solutions.