What an inequality is, and how to solve and graph inequalities

The speed limit sign says 65, your child's homework says "solve 3x − 4 > 11", and both lead to the same question: what is an inequality? An inequality in math is a statement that two values are not equal, and its answer is usually a whole range of numbers instead of one. If you can solve an equation, you already know how to solve inequalities with the same moves plus one rule about negatives, and how to graph inequalities comes down to a boundary, a circle and an arrow.

What is an inequality?

An inequality is a math statement that compares two values with one of five signs: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to) and ≠ (not equal to). The pointed end of < or > faces the smaller value, so 3 < 8 and 8 > 3 say the same thing.

An equation such as x + 2 = 7 has one answer, x = 5. An inequality such as x + 2 > 7 has infinitely many: 6, 5.5, 20 and 1,000 all work, and 5 does not. OpenStax Elementary Algebra 2e puts it this way for x > 3: "Any number greater than 3 is a solution to the inequality."

Grade 6 is where inequalities start in the US. Common Core standard 6.EE.B.8 asks students to "write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem" and to "recognize that inequalities of the form x > c or x < c have infinitely many solutions". The same standard asks them to show those solutions on number line diagrams.

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Real-life inequalities: speed limits, ages and "at least"

Real-life inequalities are limits and minimums written as math. A 65 mph speed limit says your speed s must satisfy s ≤ 65, because 65 itself is allowed. A ride that requires riders to be at least 48 inches tall says h ≥ 48, and a meal plan with "at least 3 servings of vegetables" says v ≥ 3.

The phrase decides the sign. OpenStax notes that "the phrase 'at least' is the same as 'greater than or equal to'", so "at least" means ≥ and "at most" or "no more than" means ≤. "More than" and "fewer than" leave the boundary out, so they become > and <.

How to solve an inequality, step by step

Solving an inequality means finding every value of the variable that makes it true, and the method matches the one for solving for x in an equation. Take 3x − 4 > 11.

1. Read the sign. The sign is >, "greater than", so the answer will be "x is greater than some number" unless a negative divisor flips it.
2. Undo addition or subtraction. Add 4 to both sides: 3x − 4 + 4 > 11 + 4, which gives 3x > 15. Adding or subtracting never changes the sign.
3. Undo multiplication or division. Divide both sides by 3: x > 5. Dividing by a positive number keeps the sign as it is.
4. Check a number that should work. Try x = 6: 3 × 6 − 4 = 14, and 14 > 11 is true.
5. Check the boundary. Try x = 5: 3 × 5 − 4 = 11, and 11 > 11 is false, so 5 is not a solution. That result tells you to draw an open circle at 5.
6. Write and graph the answer. The solution is x > 5: every number greater than 5, shown with an open circle at 5 and an arrow to the right.

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When do you flip the inequality sign?

The inequality sign flips when you multiply or divide both sides by a negative number, and at no other time. OpenStax states the rule in one line: "When we divide or multiply an inequality by a negative number, the inequality sign reverses." Adding, subtracting, and multiplying or dividing by a positive number leave the sign alone.

A true statement shows why the rule exists. The statement 2 < 5 is true, but multiplying both sides by −1 gives −2 and −5, and −2 is greater than −5 on the number line. Keeping the sign would give −2 < −5, which is false, so the sign must turn around to −2 > −5.

Take −2x ≤ 8: divide both sides by −2 and flip ≤ to ≥ to get x ≥ −4. The boundary belongs in the answer, because −2 × (−4) = 8 and 8 ≤ 8 is true.

A number below −4 fails: x = −5 gives −2 × (−5) = 10, and 10 ≤ 8 is false, so the answer points right. If multiplying negatives feels shaky, the guide to negative numbers covers the sign rules.

Inequality signs compared: words, circles and notation

An inequality sign comes with its own words, its own circle on a number line and its own interval notation. Interval notation is optional in grades 6 to 8, but algebra textbooks use it.

Interval notation in one rule

Interval notation writes a solution set as its two ends, with a parenthesis for an end that is left out and a bracket for an end that is included. OpenStax explains that "the open parentheses symbol, (, shows that the endpoint of the inequality is not included" and "the open bracket symbol, [, shows that the endpoint is included". Infinity (∞) always gets a parenthesis, because no number reaches it.

The circle and the bracket carry the same message. An open circle matches a parenthesis and a closed circle matches a bracket, so x ≥ 2 is a closed circle at 2 and the interval [2, ∞).

How to graph inequalities on a number line

Graphing an inequality on a number line takes three decisions: where the boundary goes, which circle to draw, and which way to shade. For x > 2, the boundary is 2, the circle is open because 2 is not a solution, and the arrow points right toward the bigger numbers.

The circle comes from the sign. Use an open circle for < and >, and a closed (filled) circle for ≤ and ≥. Some books, OpenStax among them, draw a parenthesis or a bracket at the boundary in place of a circle, and the meaning is the same.

The arrow comes from the direction. With x on the left, > and ≥ shade to the right and < and ≤ shade to the left. Pick one number on the shaded side and test it: for x ≤ −1, try x = −3, and −3 ≤ −1 is true.

Rule: rewrite with x on the left before you shade

An inequality with the variable on the right, such as 7 < x, is easy to shade the wrong way. OpenStax gives the fix: "x > a has the same meaning as a < x". Read 7 < x from the x side, "x is greater than 7", and write x > 7 before you draw.

The same move tidies up a solved answer. Solving 10 ≥ 2x + 4 gives 6 ≥ 2x and then 3 ≥ x, which you rewrite as x ≤ 3: a closed circle at 3 and an arrow left, and x = 3 checks out because 2 × 3 + 4 = 10. A number line is one axis of a coordinate grid, and the guide to the coordinate plane shows how the same idea grows into two dimensions.

How to check your progress with inequalities

Five short problems show whether you understand inequalities or are following a pattern. Solve each one without notes, check the boundary, then compare.

• One step: solve x + 7 ≤ 12. Subtracting 7 gives x ≤ 5, a closed circle at 5 and an arrow left.
• Positive divisor: solve 4x > −12. Dividing by positive 4 keeps the sign, so x > −3.
• Negative divisor: solve −3x > 12. Dividing by −3 flips the sign, so x < −4. Check x = −5: 15 > 12 is true.
• Hidden negative: solve 5 − x < 2. Subtracting 5 gives −x < −3, and dividing by −1 gives x > 3.
• Word problem: you earn $50 a week plus $3 per sale and want at least $100. That gives 50 + 3s ≥ 100, so 3s ≥ 50 and s ≥ 50 ÷ 3, about 16.7, which means at least 17 sales.

A workable test: the boundary check

The boundary check is a one-minute test that catches a wrong circle and a wrong direction. Put the boundary number into the original inequality: it should make both sides equal, and the sign tells you whether that counts. Then test one number on the shaded side, which must give a true statement.

The word problem comes straight from the curriculum. Common Core 7.EE.B.4b asks grade 7 students to "solve word problems leading to inequalities of the form px + q > r or px + q < r" and uses the same salesperson example. Check it: 17 sales pay 50 + 51 = $101, and 16 sales pay 50 + 48 = $98, which falls short.

Common inequality mistakes and edge cases

Four mistakes cause most wrong answers on inequality homework, and each has a quick fix.

• Forgetting to flip: dividing −2x ≤ 8 by −2 and writing x ≤ −4 gives the opposite half of the line. Say "negative, flip" out loud every time you divide by a negative.
• Flipping for no reason: a negative number in the answer does not flip anything. In 2x > −8, you divide by positive 2, so x > −4 keeps its sign.
• Wrong circle: a closed circle on x > 5 claims 5 works when it does not. Check the boundary by substitution before you fill a circle.
• Wrong direction with x on the right: 7 < x shades right, because it means x > 7. Rewrite with x on the left first.

Edge case: compound inequalities, no solution and all real numbers

A compound inequality such as −2 < x ≤ 3 gives the variable two limits at once. OpenStax Intermediate Algebra 2e calls it a double inequality and explains that "a < x < b is equivalent to a < x and x < b". On the number line, draw an open circle at −2, a closed circle at 3 and shade the segment between them; in interval notation it is (−2, 3].

Solving x + 1 < x gives 1 < 0, a false statement, so the inequality has no solution. Solving x + 3 > x gives 3 > 0, which is always true, so every real number is a solution. OpenStax calls the false case a "contradiction" and the always-true case an "identity".

Your inequalities checklist and next steps

• Sign: read the sign out loud and translate "at least" to ≥ and "at most" to ≤.
• Same moves: undo addition and subtraction, then multiplication and division, on both sides.
• Negative rule: flip the sign only when you multiply or divide by a negative number.
• Boundary check: substitute the boundary and one shaded number into the original inequality.
• Graph: put x on the left, choose an open or closed circle, and shade toward the numbers that work.
• Order: simplify each side with the order of operations before you move any terms.

The free solve for x game practices the inverse operations behind every inequality, right in the browser with no sign-up. Adults rebuilding algebra can follow the plan in how to get better at math as an adult or try the math game for adults.

MathIt is a free app for iOS and Android with about 90 game types and adaptive difficulty, so a few minutes of daily practice keeps negative numbers and equation steps quick.