An inequality is solved almost exactly like an equation, with the same inverse operations. The one difference: multiplying or dividing by a negative number reverses the inequality sign. An inequality usually has infinitely many solutions, so the answer is a region, drawn on a number line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it reverses
3 < 5 is true. Multiply both sides by −1: −3 and −5, and −3 is greater than −5. The order genuinely flips. Adding or subtracting never flips it: 3 − 10 < 5 − 10 is still true.
A two-step one
5 − 3x ≤ 14: subtract 5, −3x ≤ 9, divide by −3 and flip, x ≥ −3. Check x = 0: 5 − 0 = 5 ≤ 14, true, and 0 is in the region x ≥ −3.
Avoiding the flip
5 − 3x ≤ 14 can also be solved by adding 3x to both sides: 5 ≤ 14 + 3x, then −9 ≤ 3x, −3 ≤ x. The same answer with no negative division, so no flip to remember.
Graphing
An open circle excludes the endpoint, for < and >. A closed circle includes it, for ≤ and ≥. Shade toward the side the variable is allowed: x < −4 shades left from an open circle at −4.
Compound inequalities
"And": −2 < x ≤ 3 needs both parts, the overlap between −2 and 3. "Or": x < −1 or x > 4 needs either, two separate regions. Solve each part, then combine as the word says.
Test a value
After solving, test one value from the answer region in the original inequality. x = −5 in −2x > 8 gives 10 > 8, true. A missed flip would have given x > −4, and x = 0 would fail: 0 > 8 is false.
Step 2: Try It Yourself
Tap and try it out.
Solve −3 ≤ 5 − 2x < 11 and describe the region
- 1−3 ≤ 5 − 2x < 11as given: an "and" inequality
Step 0 of 4
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Step 3: In Real Life
Rules with limits
Applicants must be at least 18: a ≥ 18. A trailer’s load must not exceed 2,000 pounds: w ≤ 2,000. Dividing by a negative flips the sign, which matters when the rate is a loss.
Step 4: Watch an Example
One step at a time.
Watch Marcus Solve −2x > 8
Marcus must find every value of x that works.
- Step 1
The x is multiplied by −2, so he divides both sides by −2. Dividing by a negative reverses the sign, so > becomes <: x < −4.
Step 5: Your Turn
Practice makes it stick.
The Solve
Problem 1 of 2
3x < 12. What is the largest whole number x can be?
The Budget
Problem 2 of 2
Items cost $7 each and you have $50. What is the most you can buy?
Solve and Sketch
1 of 8
x + 4 < 9. What is the boundary value for x?
2 of 8
2x ≥ 14. What is the boundary value?
3 of 8
−3x > 9. What is the boundary value?
4 of 8
Does dividing by −4 reverse the sign?
5 of 8
Does subtracting 6 reverse the sign?
6 of 8
5x + 3 ≤ 23. What is the boundary value?
7 of 8
Sort each symbol by the circle it uses on a number line.
Tap something to move it.
- Empty
- Empty
8 of 8
−2x ≤ 10. What is the boundary value?
Step 6: Quick Check
Show what you know.
Question 1 of 2
4x > 20. What is the boundary value for x?
Question 2 of 2
When does an inequality sign reverse?
What You Learned
- Solve an inequality like an equation, with one exception.
- Multiplying or dividing by a negative reverses the sign; moving the x term instead avoids it.
- Open circles exclude the endpoint; closed circles include it.
- Test one value from the answer region in the original inequality.