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:
Symbol | Reads as | Example | Includes the boundary? |
|---|---|---|---|
$<$ | is less than | $x < 5$ | No |
$>$ | is greater than | $x > 5$ | No |
$\leq$ | is less than or equal to | $x \leq 5$ | Yes |
$\geq$ | is greater than or equal to | $x \geq 5$ | 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 $-3 < x \leq 4$ (a range) or $x < -1$ or $x > 5$ (two pieces).
Polynomial and rational inequalities — the variable appears squared or higher, or in a denominator: $x^2 - 4 > 0$. 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.
Simplify each side. Distribute, combine like terms.
Isolate the variable term. Add or subtract to gather the variable on one side.
Divide or multiply to finish. If that final number is negative, flip the symbol.
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 $x + 5 \leq 8$.
Subtract 5 from both sides. No multiplication by a negative, so no flip:
$$x \leq 3$$
Final answer: $x \leq 3$. Interval notation: $(-\infty, 3]$ — the bracket because $\leq$ includes 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 $-4(0) = 0 < -16$? No — $0$ is not less than $-16$. The "solution" fails the original inequality, so the direction must be wrong.
Why it breaks. Dividing by a negative reflects both sides across zero, which reverses their order. Keeping the symbol leaves the statement false.
The correct way. Divide both sides by $-4$ and flip the symbol:
$$x > 4$$
Test $x = 5$: $-4(5) = -20 < -16$. True. The flipped direction is the one that works.
Final answer: $x > 4$. Interval notation: $(4, \infty)$.
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 $(6, \infty)$.
Example 4
Solve $\dfrac{x}{-2} + 1 \geq 4$.
Subtract 1 from both sides:
$$\frac{x}{-2} \geq 3$$
Multiply both sides by $-2$ — negative, so flip $\geq$ to $\leq$:
$$x \leq -6$$
Final answer: $x \leq -6$, or $(-\infty, -6]$. The flip is triggered by the multiply-by-negative, not by the negative already sitting in the problem.
Example 5
Solve the compound inequality $-3 < 2x + 1 \leq 7$.
Work on all three parts at once. Subtract 1 throughout:
$$-4 < 2x \leq 6$$
Divide every part by 2 — positive, no flip:
$$-2 < x \leq 3$$
Final answer: $-2 < x \leq 3$, or $(-2, 3]$. The solution set is the band between $-2$ (open) and $3$ (closed).
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 $\geq 90$, applied to the average:
$$\frac{85 + s}{2} \geq 90$$
Multiply both sides by 2 — positive, no flip:
$$85 + s \geq 180$$
Subtract 85:
$$s \geq 95$$
Final answer: $s \geq 95$, or $[95, \infty)$ capped by the test's maximum. The phrase "at least" is the translation key — it always becomes $\geq$.
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 $>$. On the line, a hollow dot; in interval notation, a parenthesis: $x > 4$ is $(4, \infty)$.
Closed dot / bracket — the boundary is included. Use for $\leq$ and $\geq$. On the line, a filled dot; in interval notation, a bracket: $x \leq 3$ is $(-\infty, 3]$.
Infinity always takes a parenthesis. You can never "reach" infinity, so it's never bracketed: $[95, \infty)$, never $[95, \infty]$.
Write smaller number first. Interval notation reads left-to-right like the number line: $(-2, 3]$, not $(3, -2]$.
For a deeper treatment of bracket-versus-parenthesis edge cases and unbounded sets, the interval notation guide carries the full reference.
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. "Spend no more than $50," "score at least 90 to pass," "the bridge holds at most 10 tonnes" — each is an inequality, and the solution is the safe range, not a single value.
Engineering tolerances. A part machined to "within 0.01 mm" is a compound inequality on its dimension. Staying inside the range is what keeps machines working.
Optimisation and linear programming. Whole industries — airline scheduling, factory output, delivery routing — are solved by maximising something subject to a stack of inequality constraints. The number-line region you shade here becomes a feasible region in two and three dimensions later.
The reason inequalities are taught right after equations is that the world hands out ranges far more often than exact values. A speed limit is $\leq 60$, not $= 60$. A medicine dose is a safe window, not a single number. Equations describe the special case; inequalities describe the rule.
Where Inequalities Go Sideways
Mistake 1: Forgetting to flip the symbol on a negative
Where it slips in: The last step divides or multiplies both sides by a negative number, and the student leaves the symbol pointing the same way.
Don't do this: Solve $-2x > 6$ as $x > -3$. Test $x = 0$: $-2(0) = 0 > 6$ is false, so the direction is wrong.
The correct way: Flip on the negative. $-2x > 6$ becomes $x < -3$. Then test a value from your solution set against the original to confirm the direction.
Mistake 2: Using the wrong dot or bracket on the boundary
Where it slips in: Graphing or writing $x \leq 3$, a student uses an open circle or a parenthesis.
Don't do this: Treat $\leq$ and $<$ as interchangeable on the graph. The "or equal to" means the boundary is part of the solution.
The correct way: $\leq$ and $\geq$ get a closed dot and a bracket; $<$ and $>$ get an open circle and a parenthesis. The second-guesser who keeps swapping them should read the symbol aloud — "less than or equal to" — and let the words pick the bracket.
Mistake 3: Reversing the order in interval notation
Where it slips in: Writing the interval for $-2 < x \leq 3$ as $(3, -2]$.
Don't do this: Write the larger number first. Interval notation follows the number line, smaller value on the left.
The correct way: $(-2, 3]$ — smaller bound, then larger, with the bracket type matching each end's symbol.
Key Takeaways
An inequality compares expressions with $<$, $>$, $\leq$, or $\geq$ and has a range of solutions, not a single value.
Solve like an equation — but flip the symbol whenever you multiply or divide both sides by a negative.
$\leq$ and $\geq$ include the boundary (closed dot, bracket); $<$ and $>$ exclude it (open circle, parenthesis).
Interval notation reads left to right, smaller bound first, with infinity always in a parenthesis.
Testing one value from your solution set against the original inequality catches a forgotten flip instantly.
Practice These Problems
Solve and write in interval notation: $2x - 3 \leq 9$.
Solve and graph: $-5x > 20$. (Mind the flip.)
Solve the compound inequality: $-1 \leq 3x + 2 < 11$.
A taxi charges $4 plus $2 per mile, and you have at most $30. Write and solve the inequality for the miles $m$ you can travel.
Answer to Question 1: $x \leq 6$, or $(-\infty, 6]$. Answer to Question 2: $x < -4$, or $(-\infty, -4)$. Answer to Question 3: $-1 \leq x < 3$, or $[-1, 3)$. Answer to Question 4: $2m + 4 \leq 30$, so $m \leq 13$ miles.
If Question 2 gave you $x > -4$, return to Mistake 1 and flip on the negative.
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