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When to Flip the Inequality Sign on the SAT

Solving an inequality is nearly identical to solving an equation, with one exception that the SAT tests on purpose. Multiplying or dividing both sides by a negative number reverses the direction of the sign. Nothing else does, and knowing exactly where the exception applies is most of the skill.

The rule

Reverse the inequality sign only when you multiply or divide both sides by a negative number.

Here is why it happens. The statement 3 is less than 7 is true. Multiply both sides by negative 1 and you get negative 3 and negative 7. Since negative 3 sits to the right of negative 7 on the number line, the true statement is now negative 3 is greater than negative 7. Multiplying by a negative mirrors the number line, so the order reverses.

Work through negative 4x plus 6 is greater than 26. Subtract 6 from both sides, which is addition and changes nothing about the direction, to get negative 4x is greater than 20. Now divide both sides by negative 4, and that is the step that flips: x is less than negative 5. Check with a test value. If x equals negative 10, then negative 4 times negative 10 plus 6 is 46, which is indeed greater than 26.

The rule does not fire when you merely add or subtract a negative number. In x minus 9 is less than 2, adding 9 to both sides gives x is less than 11 with the sign untouched. Students who flip here are reacting to the sight of a negative number rather than to the operation being performed. Ask yourself one question at each step: am I multiplying or dividing by a negative.

A related pattern appears with variables on both sides. From 5 minus 2x is greater than or equal to 11, subtract 5 to get negative 2x is greater than or equal to 6, then divide by negative 2 and flip: x is less than or equal to negative 3. An alternative that avoids the flip entirely is to move the variable to the other side instead: 5 is greater than or equal to 11 plus 2x, so negative 6 is greater than or equal to 2x, so negative 3 is greater than or equal to x, which says the same thing.

Finish by testing one number from your solution set in the original inequality. It takes ten seconds and it catches a flipped or unflipped sign immediately, which matters because the answer choices on these questions typically include both directions.

The trap

Flipping the sign after adding or subtracting a negative number, which is not an operation that reverses anything.

Try it

Try one
Real question

Solve -(x - 5) > 2.

A greenhouse must keep its relative humidity above 58 percent and at or below 76 percent. The humidity, in percent, is modeled by 34 + 3m, where m is the number of misting cycles run per hour. What is the greatest number of misting cycles per hour allowed?

  • A) 6
  • B) 8
  • C) 14 ✓
  • D) 15

Why: Set 34 + 3m <= 76: 3m <= 42, so m <= 14. Answering 6 means you divided the range between the two limits by 3. Answering 8 means you used the 58 percent lower limit instead of the upper limit. Answering 15 means you took one more cycle, which puts the humidity at 79 percent, above the limit.

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Questions students ask

Do I flip the sign when I take the reciprocal of both sides?

For positive quantities, yes, taking reciprocals reverses the order, since one half is less than one third is false while 2 is less than 3 is true. SAT linear inequality questions rarely require this, so focus on the multiply-or-divide-by-a-negative rule and test a value if a reciprocal appears.

How do I check my answer to an inequality?

Pick any number inside your solution set and substitute it into the original inequality. If the statement is true, the direction is right. Then try a number outside the set and confirm the statement becomes false.

Guides that cover this

More on linear inequalities