The key idea
Adding, subtracting, and multiplying or dividing by a positive number all keep an inequality true. Multiplying or dividing by a negative number reverses the order of numbers on the number line. That is why the sign has to flip.
So the rule in action looks like this:
| Symbol | Means | Circle on the number line | Interval notation (for a = 2) |
|---|---|---|---|
| less than 2 | open circle at 2, shade left | ||
| greater than 2 | open circle at 2, shade right | ||
| less than or equal to 2 | closed circle at 2, shade left | ||
| greater than or equal to 2 | closed circle at 2, shade right |
Worked examples
Example 1: a two-step inequality
Problem Solve and describe its graph.
- Add 5 to both sides.
- Divide both sides by 3. It is positive, so the sign stays the same.
- Graph: a closed circle at 6, because 6 itself works, and shade everything to the left.
Answer
Example 2: dividing by a negative
Problem Solve .
- Subtract 7 from both sides. Subtracting never flips the sign.
- Divide both sides by . This is a negative number, so flip to .
- Test a point to be sure. is in the answer, and , which is greater than 19. It works.
Answer
Example 3: a compound "and" inequality
Problem Solve .
- This means is between and 11. Do every step to all three parts. Subtract 5 everywhere.
- Divide all three parts by 2.
- Graph: open circle at , closed circle at 3, and shade the segment between them.
Answer
Example 4 (test-hard): a word problem with variables on both sides
Problem Gym A charges a $40 joining fee plus $25 a month. Gym B has no fee and charges $35 a month. After how many whole months is Gym A cheaper?
- Let be the number of months. Gym A is cheaper when its total is less than Gym B's.
- Subtract from both sides. Keeping the variable on the right avoids a negative coefficient.
- Divide by 10 (positive, so no flip).
- Read it as . At exactly 4 months both gyms cost $140, so Gym A is not cheaper yet. At 5 months Gym A costs $165 and Gym B costs $175.
Answer : Gym A is cheaper from month 5 on.
Common mistakes
- Forgetting to flip when dividing by a negative. becomes , not . Fix: every time you divide, look at the sign of the number first.
- Flipping when you should not. Subtracting a negative, or having a negative answer, does not flip anything. Only multiplying or dividing by a negative does.
- Mixing up open and closed circles. The line under and means "or equal," so that endpoint is included and gets a filled circle.
- Doing a step to only one part of a compound inequality. In , subtract 5 from all three parts, not just the middle.
- Reading 40 < 10m backwards. and say the same thing. Rewriting it with the variable first makes the graph easier to draw.
Quick methods
Practice
5 practice questions
Solve .
Show answer
Answer:
Subtract 2: . Divide by and flip: . The choice forgets the flip.
Which value of x is a solution of ?
- 6
- 7
- 8
- 9
Show answer
Answer: 6
Distribute and solve: , so and . Only 6 is less than 7. The value 7 makes both sides equal (20 and 20), and that fails a strict .
Solve .
Show answer
Answer:
Multiply all three parts by 2: . Subtract 3 from all three parts: . The choice stops one step early.
Which graph shows the solution of or ?
- Open circle at shading left, and closed circle at 5 shading right
- Closed circle at shading left, and open circle at 5 shading right
- Open circle at and closed circle at 5, shading between them
- Open circle at shading left, and closed circle at 5 shading right
Show answer
Answer: Open circle at shading left, and closed circle at 5 shading right
The first part gives (open circle, strict), and the second gives (closed circle). "Or" means either piece works, so there are two separate rays. Shading between them would be an "and" answer, and no number is both less than and at least 5.
Sam has $50 and saves $15 each week. Which inequality gives the number of weeks until he has at least $200?
Show answer
Answer:
"At least" means : , so and . The choice comes from adding the 50 instead of subtracting it. leaves out week 10, when he has exactly $200.
Frequently asked questions
Why do you flip the inequality sign when dividing by a negative?
Multiplying or dividing by a negative mirrors every number across zero. Bigger numbers become smaller ones: 2 is less than 5, but is greater than . Flipping the sign keeps the statement true after that mirror.
What is the difference between "and" and "or" inequalities?
An "and" inequality needs both parts true at once, so the answer is the overlap, usually one segment like . An "or" inequality needs at least one part true, so the answer is everything from both parts, often two rays pointing away from each other.
When do I use an open circle and when a closed circle?
Use a closed (filled) circle when the endpoint is part of the answer, which happens with and . Use an open circle with and , because the endpoint itself does not work.
Do I ever flip the sign when I move a term to the other side?
No. Moving a term is really adding or subtracting it on both sides, and that never flips the sign. Only multiplying or dividing both sides by a negative number does. Swapping the two sides, like rewriting as , also turns the symbol around, because you are reading it from the other end.
