MOSTMATHHome

Chapter 7 · Inequalities · page 35 of 68

Dividing by a negative flips the sign

levelAlgebra 1
standard7.EE.B.4b
practiceInequalities

Multiplying or dividing both sides of an inequality by a negative number reverses the order of everything on the number line, so the sign has to flip to stay true. 2 < 5 is true, and −2 > −5 is true. If you must divide by −3, the < becomes >.

How to use it

  1. Whenever you multiply or divide by a negative, flip the sign in the same step.
  2. Alternative: add the variable term to the other side so its coefficient is positive, and never divide by a negative at all.
  3. Check with a number from your answer range in the original inequality.

Example

−2x+3>9⇒−2x>6⇒x<−3-2x + 3 > 9 \Rightarrow -2x > 6 \Rightarrow x < -3
−3(y−2)<2y+26⇒−3y+6<2y+26⇒−5y<20⇒y>−4-3(y - 2) < 2y + 26 \Rightarrow -3y + 6 < 2y + 26 \Rightarrow -5y < 20 \Rightarrow y > -4
Or: 6−26<2y+3y⇒−20<5y⇒−4<y\text{Or: } 6 - 26 < 2y + 3y \Rightarrow -20 < 5y \Rightarrow -4 < y

Where it goes wrong

  • Flipping when subtracting a negative number. Subtraction never flips.
  • Flipping at the wrong step, before the division happens.
  • Not flipping at all. Test it: x = 0 makes −2x + 3 > 9 false, so x > −3 cannot be right.

Related

Solving an inequalityWhen the x term is negativeMultiplying and dividing signs
Practice this