Solving Inequalities — Solve a simple inequality and...

SECTION 1

Where Did Inequalities Come From?

Equations use an equal sign to say two things are the same. But what happens when things are not equal? In real life, we deal with "less than" and "greater than" all the time. You might need to score more than 80 points to pass a test, or spend less than $20 at the store. These situations led mathematicians to create inequalities (math statements that compare values using symbols like < and >).

~300 BCE

Ancient Greek Comparisons

Euclid and other Greek mathematicians compared lengths and areas, writing out phrases like "this is greater than that" in words. They had no symbols yet.

1631

The Symbols Are Born

English mathematician Thomas Harriot introduced the < (less than) and > (greater than) symbols. These are the same symbols you use today!

1734

Adding the "Or Equal To" Bar

French mathematician Pierre Bouguer added a line under the < and > symbols to create ≤ (less than or equal to) and ≥ (greater than or equal to).

1900s

Inequalities in Modern Math

Inequalities became essential in algebra, economics, and computer science. Today they appear on tests like the SHSAT and are used to solve real-world problems.

So here is the big question: when a problem says x + 3 > 7, how do you figure out which values of x make it true? That is exactly what this lesson will teach you.

SECTION 2

Core Principles of Inequalities

An inequality is a math sentence that uses a comparison symbol instead of an equal sign. Instead of one answer, an inequality usually has many values that make it true. Let's look at the key ideas you need.

The Four Inequality Symbols

Use < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare two expressions.

Solving Is Like Equations — Almost

You solve inequalities the same way you solve equations: add, subtract, multiply, or divide both sides. But there is one important extra rule (see card 3).

Flip When You Multiply or Divide by a Negative

If you multiply or divide both sides by a negative number, you must flip the inequality symbol. This is the biggest rule students forget!

The Solution Is a Range

An equation like x = 5 has one answer. An inequality like x > 3 has infinitely many answers: 4, 3.5, 100, and so on. We show this range on a number line.

Open vs. Closed Circles

On a number line, an open circle (○) means "not included" (for < or >). A closed circle (●) means "included" (for ≤ or ≥).

✦ KEY TAKEAWAY

Think of an inequality like a velvet rope at a theme park ride. A sign that says "You must be at least 48 inches tall" is really saying height ≥ 48. Everyone 48 inches or taller gets in. That is a whole range of values, not just one number.

SECTION 3

Seeing Inequalities on the Number Line

A number line graph is the best way to picture an inequality's solution. It shows every value that makes the inequality true. The diagram below compares the four types of simple inequalities.

Each number line shows the solution to a different inequality. Notice how open circles mean the boundary number is NOT included, while closed (filled) circles mean it IS included. The arrow shows the direction of all the values that work.

Here is a quick way to remember: if the symbol has a line under it (≤ or ≥), the circle is filled in. If there is no line (< or >), the circle is open. The arrow always points toward the values that make the statement true.

SECTION 4

The Math Behind Solving Inequalities

Solving an inequality means getting the variable alone on one side. You follow the same steps you already know from solving equations. Here are the key rules written as formulas.

ADDITION / SUBTRACTION RULE

If a < b, then a + c < b + c and a − c < b − c

You can add or subtract the same number on both sides. The inequality symbol stays the same.

MULTIPLICATION / DIVISION BY A POSITIVE

If a < b and c > 0, then a × c < b × c and a ÷ c < b ÷ c

Multiplying or dividing both sides by a positive number keeps the symbol the same.

MULTIPLICATION / DIVISION BY A NEGATIVE — THE FLIP RULE

If a < b and c < 0, then a × c > b × c and a ÷ c > b ÷ c

Multiplying or dividing both sides by a negative number means you must flip the inequality symbol. For example, < becomes > and ≥ becomes ≤.

💡 Why Does the Symbol Flip?

Think about it: 2 < 5 is true. Now multiply both sides by −1. You get −2 and −5. On a number line, −2 is to the right of −5, so −2 > −5. The order reversed! That is why you flip the symbol whenever you multiply or divide by a negative.

SECTION 5

Step-by-Step Solving Process

Let's break the solving process into clear steps. The flowchart below shows how to approach any simple inequality.

Follow these four steps every time. The most important reminder is at Step 3: if you multiply or divide by a negative number, you must flip the inequality symbol.

After you solve, always check your answer by picking a number from your solution set and plugging it back in. If the original inequality is true, you are correct.

SECTION 6

Worked Example: Solving and Graphing

Let's solve the inequality −3x + 7 ≤ 1 step by step and then graph the solution on a number line.

Solve −3x + 7 ≤ 1

Step 1 — Read the inequalityWe need to find all values of x that make −3x + 7 ≤ 1 true. Our goal is to get x alone on one side.

Step 2 — Subtract 7 from both sidesWe want to move the +7 away from the variable term. Subtract 7 from both sides: −3x + 7 − 7 ≤ 1 − 7 −3x ≤ −6

−3x ≤ −6

Step 3 — Divide both sides by −3 (FLIP the symbol!)We divide by −3 to get x alone. Because we are dividing by a negative number, we must flip ≤ to ≥. −3x ÷ (−3) ≥ −6 ÷ (−3) x ≥ 2

x ≥ 2

Step 4 — Interpret the solutionThe solution x ≥ 2 means every number that is 2 or greater works. On a number line, draw a closed circle at 2 (because 2 is included) and shade to the right.

Step 5 — Check with a test valuePick x = 5 (which is ≥ 2). Plug it in: −3(5) + 7 = −15 + 7 = −8. Is −8 ≤ 1? Yes! ✓ Our answer is correct.

SECTION 7

Common Mistakes and How to Avoid Them

Many students lose points on the SHSAT because of a few common mistakes. Knowing these traps ahead of time can save you.

Watch out for these four common traps on the SHSAT

Mistake	Why It's Wrong	How to Fix It
Forgetting to flip the symbol when dividing by a negative	Dividing by a negative reverses the order of numbers. If you don't flip, your answer set is backwards.	Circle the negative sign in your work. Write "FLIP!" next to it as a reminder.
Using a closed circle when the symbol is < or >	A closed circle means the boundary value is included, but < and > mean it is NOT included.	Check for the "or equal to" bar under the symbol. No bar = open circle.
Drawing the arrow in the wrong direction	Shading the wrong side means you are showing values that make the inequality false.	After solving, plug in a value from your shaded region to verify.
Flipping the symbol when adding or subtracting	The flip rule ONLY applies when multiplying or dividing by a negative. Addition and subtraction never change the symbol.	Ask yourself: Am I multiplying or dividing by a negative? If not, keep the symbol the same.

✦ KEY TAKEAWAY

Think of the flip rule like reversing a camera image. When you look in a mirror (multiply by −1), left and right swap. In the same way, when you multiply or divide an inequality by a negative, the direction of the comparison swaps.

SECTION 8

Connecting to More Advanced Inequality Topics

The simple inequalities you just learned are the building blocks for harder topics you will see in high school and beyond. Here's a quick peek at what comes next.

Simple inequalities now → advanced applications later

What You Know Now	What Comes Next
One-step inequality (x + 5 > 12)	Multi-step and compound inequalities (3 < 2x − 1 ≤ 9)
Number line graph	Graphing inequalities on the coordinate plane (shading a region)
One variable (x)	Two variables (y > 2x + 1) — systems of inequalities
Linear expressions	Quadratic inequalities (x² − 4 > 0) and absolute value inequalities

For the SHSAT, focus on mastering simple one-variable inequalities. If you can solve them quickly and graph them correctly, you will earn easy points. The skills you build here — isolating variables, remembering the flip rule, and checking answers — carry directly into every future math class.

SECTION 9

Practice Problems

Try these five problems. They go from easy to challenging. Work through each one on paper before reading the answer!

PROBLEM 1 — CONCEPTUAL

Which of the following values are in the solution set of x > 4? Choose all that apply: 3, 4, 4.5, 7, 100.

PROBLEM 2 — BASIC CALCULATION

Solve: x + 9 < 15. Write your answer and describe how it would look on a number line.

PROBLEM 3 — INTERMEDIATE

Solve: 2x − 5 ≥ 11. Then check your answer using x = 10.

PROBLEM 4 — APPLIED

Maya has $50. She buys a book for $12 and wants to buy T-shirts that cost $8 each. Write and solve an inequality to find the greatest number of T-shirts she can buy.

PROBLEM 5 — CRITICAL THINKING

Solve: −4x + 3 > 19. Then explain in your own words why the inequality symbol changed direction during your solution.

SUMMARY

Lesson Summary

An inequality compares two expressions using <, >, ≤, or ≥. To solve, isolate the variable using the same moves as equations: add, subtract, multiply, or divide both sides. The key rule to remember is the flip rule — when you multiply or divide by a negative number, reverse the inequality symbol.

Show your solution on a number line using an open circle for < or > (boundary NOT included) and a closed circle for ≤ or ≥ (boundary IS included). Always shade or draw an arrow in the direction of the values that work. Finally, check your answer by plugging a test value from your solution into the original inequality.