Solve one-variable inequalities and interpret solutions - TACHS Math

SECTION 1

Historical Context & Motivation

Have you ever seen a sign that says "You must be at least 48 inches tall to ride this ride"? That is an inequality in everyday life. Instead of looking for one exact answer, you are looking for a whole range of values that work.

People have used inequality ideas for thousands of years. Ancient builders needed walls "at least" a certain thickness. Merchants needed to earn "more than" their costs. Over time, mathematicians created symbols and rules so these ideas could be written clearly.

~300 BC • Euclid's Comparisons

The Greek mathematician Euclid compared lengths and areas, saying one was "greater" or "less" than another—an early form of inequality thinking.

1631 • Inequality Symbols Invented

Thomas Harriot introduced the symbols > (greater than) and < (less than) in his book, making it easier to write inequalities.

1734 • ≤ and ≥ Appear

French mathematician Pierre Bouguer first used ≤ (less than or equal to) and ≥ (greater than or equal to), completing the set of symbols we still use today.

Today • Inequalities Everywhere

Inequalities appear in speed limits, budgets, sports stats, and science experiments. Solving them is a key algebra skill tested on state assessments like the TACHS.

So how do we solve an inequality and figure out which values make it true? That is exactly what this lesson will teach you.

SECTION 2

Core Principles & Definitions

Before you solve inequalities, you need to know the key ideas. An inequality is like an equation, but instead of an equals sign it uses a comparison symbol. The answer is usually a range of values instead of one single number.

Inequality Symbols

The four symbols are < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). They tell you how two sides compare.

Solution Set

The solution set is every value of the variable that makes the inequality true. It is usually an infinite list of numbers.

Inverse Operations

You isolate the variable using inverse (opposite) operations—just like solving equations. Add undoes subtract; multiply undoes divide.

The Flip Rule

When you multiply or divide both sides by a negative number, you must flip (reverse) the inequality symbol. This is the biggest difference from equations.

Graphing on a Number Line

An open circle means the endpoint is NOT included (< or >). A closed (filled) circle means it IS included (≤ or ≥). Then shade the direction of solutions.

✦

KEY TAKEAWAY

Think of an inequality like a restaurant sign: "You must be 13 or older to sit in the balcony." The sign does not name every allowed age—it gives a boundary (13) and a direction (older). Solving an inequality means finding that boundary and direction.

SECTION 3

Visual Explanation — Number Line Graphs

A number line graph is the best way to show the solution to an inequality. It shows every value that makes the inequality true. Below you can see four examples—one for each inequality symbol.

The diagram shows all four inequality types for x compared to 3. Open circles (top two) mean the boundary is NOT part of the answer. Closed circles (bottom two) mean the boundary IS part of the answer. The shaded arrow shows the direction of all solutions.

💡

Quick Memory Trick

Think of the inequality symbol as a hungry alligator mouth. It always opens toward the bigger value. Also, if the symbol has a line under it (≤ or ≥), the circle on the number line is filled in because the endpoint counts.

SECTION 4

Mathematical Framework — Solving Inequalities

Solving a one-variable inequality is a lot like solving a one-variable equation. You use inverse operations to get the variable alone on one side. There is just one extra rule to remember.

ADDITION / SUBTRACTION PROPERTY

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 by a positive number keeps the symbol the same.

THE FLIP RULE — MULTIPLY / DIVIDE BY A NEGATIVE

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

Multiplying or dividing by a negative number REVERSES the inequality symbol. This is the most important rule to remember!

ℹ

Why Does the Symbol Flip?

Consider 2 < 5. Multiply both sides by −1: you get −2 and −5. On a number line, −2 is to the right of −5, so −2 > −5. Multiplying by a negative reverses the order of numbers, so you must reverse the symbol to keep the statement true.

SECTION 5

Types of Inequalities & How to Solve Each

Inequality problems on the TACHS come in different forms. Some need one step; others need two. The chart below shows you the most common types and how to handle each one.

Type	Example	What to Do
One-step (add/subtract)	x − 4 > 7	Add 4 to both sides → x > 11
One-step (multiply/divide positive)	3x ≤ 18	Divide both sides by 3 → x ≤ 6
One-step (divide by negative)	−2x > 10	Divide by −2 AND flip symbol → x < −5
Two-step	2x + 3 < 11	Subtract 3, then divide by 2 → x < 4
Variable on both sides	5x − 1 ≥ 3x + 7	Move variable terms to one side, constants to the other → x ≥ 4

Common inequality types tested on the TACHS

Follow this flowchart every time you solve an inequality. The yellow box is the key decision point: if you multiplied or divided by a negative number, flip the symbol.

SECTION 6

Worked Example

Let's solve a two-step inequality from start to finish and then graph it on a number line.

Solve −3x + 5 ≥ 14 and graph the solution.

1
Step 1 — Write the inequality

We start with −3x + 5 ≥ 14. Our goal is to get x alone on one side.
2
Step 2 — Undo the addition (subtract 5 from both sides)

Subtract 5 from both sides to remove the +5: −3x + 5 − 5 ≥ 14 − 5. This simplifies to −3x ≥ 9.

−3x ≥ 9
3
Step 3 — Undo the multiplication (divide both sides by −3)

Divide both sides by −3. Because −3 is negative, we must flip the inequality symbol from ≥ to ≤. So: −3x ÷ (−3) ≤ 9 ÷ (−3).

x ≤ −3
4
Step 4 — Interpret the solution

The solution x ≤ −3 means every number that is −3 or less makes the original inequality true. Examples: −3, −4, −10, −100 all work.
5
Step 5 — Graph the solution

Draw a number line. Place a closed circle at −3 (because ≤ means −3 is included). Shade the line to the left, toward smaller numbers.
6
Step 6 — Check your answer

Pick a number in the shaded region, like x = −5. Plug it in: −3(−5) + 5 = 15 + 5 = 20. Is 20 ≥ 14? Yes! The solution checks out.

✓ Verified: 20 ≥ 14 is true.

SECTION 7

Equations vs. Inequalities — Key Differences

You already know how to solve equations. Inequalities use the same basic steps, but there are some important differences. The table below helps you see what is the same and what is different.

Feature	Equations (=)	Inequalities (<, >, ≤, ≥)
Number of solutions	Usually one exact answer	Infinitely many answers (a range)
Add/Subtract both sides	Symbol stays the same (=)	Symbol stays the same
Multiply/Divide by positive	Symbol stays the same (=)	Symbol stays the same
Multiply/Divide by negative	Symbol stays the same (=)	Symbol FLIPS direction
Graphing	A single point on a number line	A ray (arrow) on a number line

Comparing equations and inequalities

✦

KEY TAKEAWAY

An equation is like asking "What exact temperature is it right now?" An inequality is like asking "What temperatures would make me need a jacket?" The answer to the inequality is a whole range of numbers, not just one.

SECTION 8

Connection to Advanced Topics

Once you master one-variable inequalities, you are ready for bigger ideas in algebra and beyond. Here is a peek at what comes next.

What You Learn Now	What Comes Next
One-variable inequalities (x > 3)	Compound inequalities (2 < x ≤ 7) — two conditions at once
Graphing on a number line	Graphing on a coordinate plane (two-variable inequalities like y > 2x + 1)
Flipping the symbol when dividing by a negative	Solving absolute value inequalities such as \|x − 4\| < 6
Interpreting solution sets	Systems of inequalities — finding where multiple conditions overlap

How today's skills build toward future algebra topics

Every one of these advanced topics uses the same basic rules you are learning now. If you nail the flip rule and the open vs. closed circle idea today, those future topics will be much easier.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

On a number line graph for x > 5, would you draw an open circle or a closed circle at 5? Explain why.

Show answer

You would draw an open circle at 5. The symbol > means "greater than" but NOT equal to. Since 5 itself is not part of the solution, the circle is left empty. A closed (filled) circle is only used with ≤ or ≥, where the boundary value is included.

Question 2: BASIC CALCULATION

Solve: x + 7 < 15

Show answer

Subtract 7 from both sides: x + 7 − 7 < 15 − 7. This gives x < 8. Since we only subtracted (not divided by a negative), the < symbol stays the same. The solution includes every number less than 8, such as 7, 0, −10, and so on. On a number line, you would draw an open circle at 8 and shade to the left.

Question 3: INTERMEDIATE

Solve: 4x − 9 ≥ 11. Graph the solution on a number line.

Show answer

First, add 9 to both sides: 4x − 9 + 9 ≥ 11 + 9, which gives 4x ≥ 20. Next, divide both sides by 4 (a positive number, so the symbol stays): x ≥ 5. The solution is x ≥ 5. On a number line, draw a closed circle at 5 (because ≥ includes 5) and shade to the right. You can check by plugging in x = 5: 4(5) − 9 = 11, and 11 ≥ 11 is true.

Question 4: APPLIED

A movie theater charges $8 per ticket. You have $50 and also need to save at least $10 for food. Write and solve an inequality to find how many tickets (t) you can buy.

Show answer

You can spend at most $50 − $10 = $40 on tickets. Each ticket costs $8, so the inequality is 8t ≤ 40. Divide both sides by 8: t ≤ 5. The solution is t ≤ 5. Since t represents a number of tickets, it must also be a whole number ≥ 0. So you can buy 0, 1, 2, 3, 4, or 5 tickets. This shows how inequalities help with real-life budgeting decisions.

Question 5: CRITICAL THINKING

Solve: −5x + 2 > 3x − 14. Then pick a number from your solution set and verify it works.

Show answer

Start by getting all x-terms on one side. Subtract 3x from both sides: −5x − 3x + 2 > −14, which gives −8x + 2 > −14. Subtract 2 from both sides: −8x > −16. Divide both sides by −8 (negative!), so flip the symbol: x < 2. The answer is x < 2. Check with x = 0: −5(0) + 2 = 2 and 3(0) − 14 = −14. Is 2 > −14? Yes! On a number line, draw an open circle at 2 and shade to the left.

SUMMARY

Lesson Summary

Summary

A one-variable inequality compares an expression to a value using the symbols <, >, ≤, or ≥. You solve it the same way you solve an equation—use inverse operations to isolate the variable. The one critical rule to remember is the flip rule: when you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

The answer to an inequality is called a solution set—a range of values, not just one number. You can graph it on a number line using an open circle (for < or >) or a closed circle (for ≤ or ≥), then shade in the direction of all solutions. Always check your answer by plugging a value from the shaded region back into the original inequality.