# How to solve and graph inequalities

Canonical: https://duckyhelper.com/learn/algebra-1/inequalities/
Updated: 2026-10-01

Solve an inequality the same way you solve an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign. The answer is a range of numbers, not one number. Graph it on a number line with an open circle for \(<\) or \(>\) and a closed circle for \(\le\) or \(\ge\), then shade the side that works.

## 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.

$$
2 < 5 \quad\text{but}\quad -2 > -5
$$

So the rule in action looks like this:

$$
-2x > 6 \;\Longrightarrow\; x < -3
$$

**Reading and graphing the four symbols**

| Symbol | Means | Circle on the number line | Interval notation (for a = 2) |
| --- | --- | --- | --- |
| \(x < 2\) | less than 2 | open circle at 2, shade left | \((-\infty, 2)\) |
| \(x > 2\) | greater than 2 | open circle at 2, shade right | \((2, \infty)\) |
| \(x \le 2\) | less than or equal to 2 | closed circle at 2, shade left | \((-\infty, 2]\) |
| \(x \ge 2\) | greater than or equal to 2 | closed circle at 2, shade right | \([2, \infty)\) |

## Worked examples

**Example 1: a two-step inequality**

Problem: Solve \(3x - 5 \le 13\) and describe its graph.

1. Add 5 to both sides.

   $$
   3x \le 18
   $$
2. Divide both sides by 3. It is positive, so the sign stays the same.

   $$
   x \le 6
   $$
3. Graph: a closed circle at 6, because 6 itself works, and shade everything to the left.

Answer: \(x \le 6\)

**Example 2: dividing by a negative**

Problem: Solve \(7 - 4x > 19\).

1. Subtract 7 from both sides. Subtracting never flips the sign.

   $$
   -4x > 12
   $$
2. Divide both sides by \(-4\). This is a negative number, so flip \(>\) to \(<\).

   $$
   x < -3
   $$
3. Test a point to be sure. \(x = -4\) is in the answer, and \(7 - 4(-4) = 23\), which is greater than 19. It works.

Answer: \(x < -3\)

**Example 3: a compound "and" inequality**

Problem: Solve \(-3 < 2x + 5 \le 11\).

1. This means \(2x + 5\) is between \(-3\) and 11. Do every step to all three parts. Subtract 5 everywhere.

   $$
   -8 < 2x \le 6
   $$
2. Divide all three parts by 2.

   $$
   -4 < x \le 3
   $$
3. Graph: open circle at \(-4\), closed circle at 3, and shade the segment between them.

Answer: \(-4 < x \le 3\)

**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?

1. Let \(m\) be the number of months. Gym A is cheaper when its total is less than Gym B's.

   $$
   40 + 25m < 35m
   $$
2. Subtract \(25m\) from both sides. Keeping the variable on the right avoids a negative coefficient.

   $$
   40 < 10m
   $$
3. Divide by 10 (positive, so no flip).

   $$
   4 < m
   $$
4. Read it as \(m > 4\). 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: \(m > 4\): Gym A is cheaper from month 5 on.

## Common mistakes

- **Forgetting to flip when dividing by a negative.** \(-4x > 12\) becomes \(x < -3\), not \(x > -3\). 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 \(\le\) and \(\ge\) 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 \(-3 < 2x + 5 \le 11\), subtract 5 from all three parts, not just the middle.
- **Reading 40 < 10m backwards.** \(4 < m\) and \(m > 4\) say the same thing. Rewriting it with the variable first makes the graph easier to draw.

## Quick methods

> **Tip: Test one point inside and one outside**
>
> Pick an easy number from your shaded region and put it into the original inequality. It should make a true statement. Then try a number outside the region; it should be false. If both are true, you forgot to flip the sign somewhere.

> **Note: Keep the variable positive**
>
> In \(40 + 25m < 35m\) we moved \(25m\) to the right instead of moving \(35m\) to the left. No negative coefficient, no flip, fewer chances to slip.

## Practice

**5 practice questions**

1. Solve \(-5x + 2 \ge 17\).
   A. \(x \ge -3\)
   B. \(x \le -3\)
   C. \(x \ge 3\)
   D. \(x \le 3\)

   Answer: \(x \le -3\). Subtract 2: \(-5x \ge 15\). Divide by \(-5\) and flip: \(x \le -3\). The choice \(x \ge -3\) forgets the flip.

2. Which value of x is a solution of \(4(x - 2) < 2x + 6\)?
   A. 6
   B. 7
   C. 8
   D. 9

   Answer: 6. Distribute and solve: \(4x - 8 < 2x + 6\), so \(2x < 14\) and \(x < 7\). Only 6 is less than 7. The value 7 makes both sides equal (20 and 20), and that fails a strict \(<\).

3. Solve \(-1 \le \frac{x + 3}{2} < 4\).
   A. \(-1 \le x < 4\)
   B. \(-5 \le x < 5\)
   C. \(-2 \le x < 8\)
   D. \(-4 \le x < 1\)

   Answer: \(-5 \le x < 5\). Multiply all three parts by 2: \(-2 \le x + 3 < 8\). Subtract 3 from all three parts: \(-5 \le x < 5\). The choice \(-2 \le x < 8\) stops one step early.

4. Which graph shows the solution of \(x - 3 < -7\) or \(2x \ge 10\)?
   A. Open circle at \(-4\) shading left, and closed circle at 5 shading right
   B. Closed circle at \(-4\) shading left, and open circle at 5 shading right
   C. Open circle at \(-4\) and closed circle at 5, shading between them
   D. Open circle at \(-10\) shading left, and closed circle at 5 shading right

   Answer: Open circle at \(-4\) shading left, and closed circle at 5 shading right. The first part gives \(x < -4\) (open circle, strict), and the second gives \(x \ge 5\) (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 \(-4\) and at least 5.

5. Sam has $50 and saves $15 each week. Which inequality gives the number of weeks \(w\) until he has at least $200?
   A. \(w \ge 10\)
   B. \(w > 10\)
   C. \(w \le 10\)
   D. \(w \ge \frac{50}{3}\)

   Answer: \(w \ge 10\). "At least" means \(\ge\): \(50 + 15w \ge 200\), so \(15w \ge 150\) and \(w \ge 10\). The choice \(\frac{50}{3}\) comes from adding the 50 instead of subtracting it. \(w > 10\) 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 \(-2\) is greater than \(-5\). 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 \(-4 < x \le 3\). 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 \(\le\) and \(\ge\). 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 \(4 < m\) as \(m > 4\), also turns the symbol around, because you are reading it from the other end.

## Related

- [How to solve multi-step equations](https://duckyhelper.com/learn/algebra-1/solving-equations/)
- [How to solve absolute value equations and inequalities](https://duckyhelper.com/learn/algebra-1/absolute-value/)
- [Linear inequalities on the SAT](https://duckyhelper.com/learn/sat-math/linear-inequalities/)
- [Algebra 1 study guides](https://duckyhelper.com/learn/algebra-1/)

## Try asking Ducky

- "I got x > -3 but the answer is x < -3. Why do I have to flip it?"
- "How do I graph x < -4 or x >= 5 on a number line?"
- "Turn this phone plan word problem into an inequality with me."

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