LEMMA

Equations & inequalities

Linear inequalities

ALEKS placementTEAS math

Solve it like an equation, but flip the inequality if you multiply or divide by a negative.

What this covers

  • Solve it the way you solve an equation: simplify, then move variable terms to one side and numbers to the other.
  • Divide by the coefficient. Flip the symbol only when that coefficient is negative.
  • Why the flip: 2 < 5 is true. Multiply both by −1 to get −2 and −5, and now −2 is the larger one.
  • Check with a number inside your answer and one outside it. Only the inside number should make the original true.
  • On a number line, an open circle means < or >, a filled circle means ≤ or ≥. Infinity always gets a parenthesis.
  • Write the answer in interval notation: smaller end first, bracket for a filled circle, parenthesis for an open one.

Worked example

Worked example

Solve -3x + 5 > 14.

  1. −3x>9Subtract 5 from both sides; the symbol does not change.
  2. x<−3Dividing by -3 reverses the inequality.
Another worked example

Solve −3x + 2 ≤ 11.

  1. −3x≤9Subtract 2 from both sides.
  2. x≥−3Divide by −3 and reverse ≤ to ≥.
  3. [−3,∞)Include −3 because the original comparison allows equality.

A common mistake

A mistake Lemma catches

From −3x>9

x>−3x<−3

“You divided both sides by -3, a negative number, but the inequality still points the same way — multiplying or dividing by a negative turns the sign around. Try it with numbers: 2 < 5, but −2 > −5.”

Try one

Sample problem

Solve the inequality.

−2x+4<−14

Practice inequalities free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this