Inequalities - Symbols, Solving, and Graphing Guide

Algebra

TL;DR
An inequality is a statement that compares two expressions with <<<, >>>, ≤, or ≥, and unlike an equation it has a range of solutions, not one. This article covers the four symbols and types, how to solve and graph inequalities on a number line, the flip-the-sign-on-negatives rule, interval notation, and the mistakes that lose marks.

What Are Inequalities?

An inequality is a mathematical statement comparing two expressions that are not necessarily equal, using one of four relation symbols. Where a linear equation has a single solution, an inequality has a set of them.

The four symbols:

The two "or equal to" symbols matter more than they look. They decide whether the boundary value itself counts — which becomes an open versus closed dot on a graph, and a parenthesis versus a bracket in interval notation.

Types of inequalities

• Linear inequalities — the variable appears to the first power: 2x+1>7. The focus of this article.
• Compound inequalities — two conditions joined, like −3<x≤4 (a range) or x<−1 or x>5 (two pieces).
• Polynomial and rational inequalities — the variable appears squared or higher, or in a denominator: x²−4>0. These build on factoring and the roots of an equation.
• Absolute-value inequalities — like |x−2|<3, which unpack into a compound inequality.

How Do You Solve an Inequality?

Solve an inequality almost exactly like a two-step equation: use inverse operations to isolate the variable. There is one rule an equation never has — multiplying or dividing both sides by a negative number flips the inequality symbol.

1. Simplify each side. Distribute, combine like terms.
2. Isolate the variable term. Add or subtract to gather the variable on one side.
3. Divide or multiply to finish. If that final number is negative, flip the symbol.
4. Write the solution set. As a number-line graph and in interval notation.

Why does the symbol flip on a negative? A real reader question, and the answer is concrete. Start with a true statement: 3<5. Multiply both sides by −1: the values become −3 and −5. But −3 is _greater_ than −5, so to keep the statement true the symbol must reverse: −3>−5. Negation reflects the number line across zero, and reflection reverses order. The flip isn't a memorised quirk; it's what keeps the inequality honest.

Examples of Inequalities

Six examples, from a one-step solve to a compound inequality and a real-world setup. The negative-coefficient flip appears where students most expect to forget it.

Example 1

Solve x+5≤8.
Subtract 5 from both sides. No multiplication by a negative, so no flip:
x≤3
Final answer: x≤3. Interval notation: (−∞,3].

Example 2

Solve −4x<−16, with the most common slip shown first.
Wrong attempt. A student divides both sides by −4 and keeps the symbol as written: x<4. Test a value the answer claims is a solution, say x=0: the original is −4(0)<−16? No — 0 is not less than −16. The "solution" fails the original inequality, so the direction must be wrong.
_The correct way._ Divide by −4 **and flip the symbol**:
x>4
Final answer: x>4. Interval notation: (4,∞).

Example 3

Solve 3x−7>11.
Add 7 to both sides:
3x>18
Divide both sides by 3 — positive, so no flip:
x>6
Final answer: x>6, or (6,∞).

Example 4

Solve x−2+1≥4.
Subtract 1 from both sides:
x−2≥3
Multiply both sides by −2 — negative, so flip ≥ to ≤:
x≤−6
Final answer: x≤−6, or (−∞,−6].

Example 5

Solve the compound inequality −3<2x+1≤7.
Work on all three parts at once. Subtract 1 throughout:
−4<2x≤6
Divide every part by 2 — positive, no flip:
−2<x≤3
Final answer: −2<x≤3, or (−2,3].

Example 6

A student needs an average of at least 90 across two tests to earn an A. The first score was 85. What must the second score, s, be?
"At least 90" is ≥90, applied to the average:
85+s/2≥90
Multiply both sides by 2 — positive, no flip:
85+s≥180
Subtract 85:
s≥95
Final answer: s≥95, or [95,∞).

Reading Solutions: Number Lines and Interval Notation

Two ways to write the same solution set, and they map onto each other exactly.

• Open circle / parenthesis — the boundary is not included. Use for <<< and >>>.
• Closed dot / bracket — the boundary is included. Use for ≤ and ≥.
• Infinity always takes a parenthesis.
• Write smaller number first. Interval notation reads left-to-right like the number line.

Why Inequalities Matter

Most real limits aren't exact — they're at least, at most, no more than. Inequalities are the language for every boundary the world actually sets.

• Constraints and budgets.
• Engineering tolerances.
• Optimisation and linear programming.

Where Inequalities Go Sideways

Mistake 1: Forgetting to flip the symbol on a negative

Mistake 2: Using the wrong dot or bracket on the boundary

Mistake 3: Reversing the order in interval notation

Key Takeaways

• An inequality compares expressions with <<<, >>>, ≤, or ≥ and has a range of solutions.
• Solve like an equation — but flip the symbol whenever you multiply or divide both sides by a negative.
• ≤ and ≥ include the boundary; <<< and >>> exclude it.
• Interval notation reads left to right, smaller bound first, with infinity always in a parenthesis.