# Inequalities - Symbols, Solving, and Graphing Guide  
[Algebra](/content/tag/algebra/index.html)

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](/content/math/algebra/linear-equations/index.html) has a single solution, an inequality has a set of them.

The four symbols:  
| Symbol | Reads as | Example | Includes the boundary? |  
| --- | --- | --- | --- |  
| <<< | is less than | x<5 | No |  
| >>> | is greater than | x>5 | No |  
| ≤ | is less than or equal to | x≤5 | Yes |  
| ≥ | is greater than or equal to | x≥5 | Yes |

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.](/content/math/algebra/roots-of-quadratic-equation/index.html)  
- **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](/content/math/algebra/two-step-equations/index.html): 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.