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: Picture It
Start with something you already know.
Having and owing
- You have $3, and your friend has $5.
- Your friend has more: 3 < 5.
- Now turn both amounts into debts: you owe $3, and your friend owes $5.
- Now you are the one better off: −3 > −5.
- Go back to $3 and $5, and hand each of you $10: $13 and $15, and your friend is still ahead.
Multiplying both sides by a negative turns having into owing (and owing into having), so the bigger amount becomes the bigger debt, and the order reverses.
Adding or taking away the same amount on both sides is like handing each person the same cash: the order stays.
The story shows multiplying by −1; dividing by any negative number reverses the order in just the same way.
The same idea with numbers: each true statement multiplied through by −1.
| Before | After × (−1) |
|---|---|
| 3 < 5 | −3 > −5 |
| 1 < 4 | −1 > −4 |
| −2 < 6 | 2 > −6 |
Every true "less than" became a true "greater than". That is the flip rule.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Which of these numbers make 10 − 2x > 4 true? Put each one in for x and see. Select every one that works.
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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
3
Step 4: 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 5: Watch an Example
One step at a time.
Watch Ana Solve 7 − 2x ≥ 1
A negative x term, so Ana knows a flip is coming and watches for the moment it happens.
- Step 1
The 7 is added to −2x, so it comes off first. Taking 7 from both sides leaves −2x ≥ −6. Nothing was multiplied or divided, so the sign stays exactly as it was.
Watch Sam Solve 1 < 2x + 3 ≤ 9
An "and" inequality: x has to make both parts true at once. Sam writes the first line, and you write the rest.
- Step 1
Three parts are held together, so every move goes to all three, the way a move on an equation goes to both sides. Taking 3 from all three parts leaves −2 < 2x ≤ 6.
Two Roads to Solve 9 − 3x > 3
The same inequality solved twice: once through the flip, once around it.
- Step 1
Road 1, through the flip. Taking 9 from both sides leaves −3x > −6. Dividing by −3, a negative, reverses the sign: x < 2. Two moves, one flip.
Step 6: Your Turn
Practice makes it stick.
The Delivery Van
Problem 1 of 17
A delivery van can carry at most 1,500 pounds. The driver weighs 180 pounds, and each box weighs 70 pounds. With the driver aboard, what is the greatest number of whole boxes the van can carry?
boxes
The Transit Card
Problem 2 of 17
Jo puts $30 on a transit card. Each ride costs $4, and she wants at least $6 left on the card. What is the greatest number of rides she can take?
rides
Solve and Sketch
Solve it like an equation. Watch for the one move that reverses the sign.
Problem 3 of 17
Solve 3x − 5 < 10. What is the boundary value of x?
Problem 4 of 17
Solve −4x ≥ 12. Which line is the answer?
Problem 5 of 17
Solve 6 − 5x > 31. Which line is the answer?
Problem 6 of 17
Solve 5x + 2 ≤ 3x + 10. What is the largest value x can take?
Problem 7 of 17
Solve 2 ≤ 3x − 4 < 11. How many whole numbers are in the region?
Whose Step?
One line in each solve is a slip. Find it, then fix it.
Problem 8 of 17
Priya solved 4 − 3x ≤ 16 and drew her answer. One line is a slip. Which?
Priya's solve of 4 − 3x ≤ 16
- 14 − 3x ≤ 16
- 2−3x ≤ 12− 4
- 3x ≤ −4÷ (−3)
- Answerclosed circle at −4, shaded to the left
Problem 9 of 17
What should Priya's line 3 read?
Problem 10 of 17
Dev solved 2x + 9 > 3 and drew his answer. One line is a slip. Which?
Dev's solve of 2x + 9 > 3
- 12x + 9 > 3
- 22x < −6− 9
- 3x < −3÷ 2
- Answeropen circle at −3, shaded to the left
Problem 11 of 17
Redo Dev’s solve of 2x + 9 > 3. Which line is the answer?
Problem 12 of 17
Maya solved −5 < 1 − 2x ≤ 7 and described her drawing. One line is a slip. Which?
Maya's solve of −5 < 1 − 2x ≤ 7
- 1−5 < 1 − 2x ≤ 7
- 2−6 < −2x ≤ 6− 1
- 33 > x ≥ −3÷ (−2)
- 4−3 ≤ x < 3
- Answeropen circle at −3, closed circle at 3, shaded between
Problem 13 of 17
How should Maya draw −3 ≤ x < 3?
Pick the Move
Decide the move before you make it.
Problem 14 of 17
12 − 4x < 4. Which first move leaves a positive x term, so no later step has to divide by a negative?
Problem 15 of 17
−4 ≤ 2 − 3x < 8. What is the right first move?
Same Solutions?
Which of these describe exactly the same numbers?
Problem 16 of 17
Select every inequality whose solutions are exactly x ≥ 2.
Problem 17 of 17
Select every description of the same region as −1 < x ≤ 4.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Find the number c that makes 7 − 2x ≥ c have exactly the solutions x ≤ 1.
Stretch 2 of 2
x < a or x > 5, where a is a whole number less than 5. Which a leaves exactly three whole numbers that are not solutions?
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.
For the grown-up
The misconception this lesson targets is that the inequality sign reverses whenever a negative is in sight: a subtraction, a negative number on the other side, a negative boundary. It reverses for one move only, multiplying or dividing both sides by a negative, because that move mirrors every number across zero and so reverses their order. Two smaller slips ride along with it: drawing a circle from the wrong sign, when each end of a region follows the sign that touches it (≤ and ≥ closed, < and > open), and changing only one part of a compound inequality, when every move goes to all three parts. Testing one number from the answer region in the original catches all three.