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Learn to turn real-life situations into inequalities, solve them, and show the answer on a number line.
People have been comparing sizes and amounts for thousands of years. Long before anyone wrote the symbols "<" or ">", traders needed to know whether they had enough goods, and builders needed to check whether a bridge was strong enough to hold weight. The math idea behind all of these questions is the inequality—a statement that one quantity is bigger or smaller than another.
Today, any time you ask "Do I have enough money?", "Can I finish in time?", or "Is this safe?", you're thinking about an inequality. In this lesson, you'll learn to take questions like those, write them as math statements, solve them, and show the answers on a number line.
Before we dive into word problems, let's make sure you're comfortable with four building blocks. If you know how equations work (like 2x + 3 = 11), you're already most of the way there—inequalities just swap the "=" for a comparison symbol.
One of the best things about inequalities is that you can see the answer. Below is a diagram showing how to graph two common types of solutions. Pay attention to the open circle versus the shading direction—those two details tell you everything.
In Example A, every number to the right of 3 makes x > 3 true. Numbers like 3.5, 4, 10, and 100 all work. But 3 itself does not work (that's why the circle is open). In Example B, every number to the left of −1 makes x < −1 true. Numbers like −2, −5, and −100 all work, but −1 itself does not.
Here's a quick memory trick: the arrow points the same direction as the inequality symbol. If the symbol opens to the right (>), you shade to the right. If it opens to the left (<), you shade to the left.
Solving an inequality of the form px + q > r or px + q < r is almost the same as solving an equation. You use inverse operations to get x by itself. There's just one extra rule you need to remember.
The trickiest part of word problems is figuring out which math symbol to use. Below is a handy table of keyword translations. Keep this in mind every time you read a word problem!
| Words You'll See | Symbol | Example Phrase → Inequality |
|---|---|---|
| more than, greater than, above, over, exceeds | > | "more than 20" → x > 20 |
| less than, fewer than, below, under | < | "fewer than 15" → x < 15 |
| at least, no less than, minimum | ≥ | "at least 50" → x ≥ 50 |
| at most, no more than, maximum | ≤ | "at most $30" → x ≤ 30 |
Now let's look at a full process diagram. This flowchart shows the steps from reading a word problem all the way to graphing the answer.
Let's walk through each of those steps with a real word problem in the next section.
4.50h + 18 > 75. This matches the form px + q > r, where p = 4.50, q = 18, and r = 75.4.50h + 18 − 18 > 75 − 18 → 4.50h > 57. Divide both sides by 4.50 to isolate h. Since 4.50 is positive, the sign stays the same: h > 57 ÷ 4.50 → h > 12.666…Inequalities are close cousins of equations, but they have some key differences. Here's a side-by-side look so you can keep them straight.
| Feature | Equation (px + q = r) | Inequality (px + q > r or < r) |
|---|---|---|
| Symbol | = | <, >, ≤, ≥ |
| Number of solutions | Usually one answer | Infinitely many (a range) |
| Solving steps | Inverse operations | Same inverse operations + flip rule |
| Graphing the answer | A single point on the number line | A ray (with open or closed circle) |
| Multiplying/dividing by a negative | No extra rule | FLIP the inequality sign! |
Mistake 1: Forgetting to flip the sign. When you divide (or multiply) both sides by a negative number, the inequality direction reverses. For example, if −2x > 10, dividing both sides by −2 gives x < −5 (not x > −5).
Mistake 2: Mixing up open and closed circles. Strict inequalities (< and >) get an open circle. "Or equal to" inequalities (≤ and ≥) get a closed (filled) circle.
Mistake 3: Shading the wrong direction. Always check your answer by plugging in a number from the shaded part. If it makes the original inequality true, you shaded correctly!
You've just learned to solve one-variable linear inequalities. That's an important building block! Here's a sneak peek at where this math takes you in the years ahead.
| What You Learned Now | What Comes Later |
|---|---|
| One inequality, one variable (px + q > r) | Compound inequalities — two inequalities joined by "and" or "or" (8th grade) |
| Graphing on a number line (1-D) | Graphing on a coordinate plane — shading regions above or below a line (Algebra 1) |
| One constraint | Systems of inequalities — multiple constraints at once, used in real-world optimization (Algebra 2) |
| Rational number coefficients | Absolute value inequalities — dealing with distance from zero (Algebra 1) |
The skills you're building right now — translating words to math, solving step by step, and interpreting answers — will carry forward into every one of those future topics. You're laying a strong foundation!
Try these five problems on your own before revealing the answers. They start easy and get more challenging. You've got this!
3x + 5 > 20. Without solving, is x = 4 part of the solution set? How about x = 6? Explain how you checked.2x + 7 < 19. Then graph the solution on a number line.½x + 3 > 8. Write your answer as an inequality and describe the solution set in words.In this lesson, you learned how to take real-world word problems and turn them into linear inequalities of the form px + q > r or px + q < r, where p, q, and r are rational numbers (fractions, decimals, or integers). You practiced four key steps: reading the problem to identify the rate, starting value, and comparison word; writing the inequality using the correct symbol (<, >, ≤, or ≥); solving algebraically by subtracting and dividing (remembering to flip the sign when dividing by a negative); and graphing on a number line with the right circle type (open for strict, closed for "or equal to") and the correct shading direction.
The solution set of an inequality is not just one number—it's an entire range of values that make the statement true. On the number line, this shows up as a ray pointing left or right. When you interpret the solution, you connect the math back to the real world: "Maya needs more than 12⅔ hours" or "Jaden can subscribe for at most 10 months." Building these translation skills now gives you a strong foundation for compound inequalities, graphing in two dimensions, and optimization problems you'll meet in later courses.