Jugami Pre-Algebra

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Inequalities

When the answer is not one number, but a whole range of them.

An equation like x + 2 = 5 has one solution. An inequality like x + 2 > 5 has many: 4, 10, 3.5 and every other number greater than 3. Inequalities describe limits: a height you must reach, a budget you can’t go over, a speed you can’t pass.

This page covers the symbols, graphing on a number line, writing inequalities from words, solving one- and two-step inequalities, and why the sign flips when you multiply or divide by a negative. Then try the practice problems and the Boundary Hunt game.

Jump to practicePlay Boundary Hunt
A number line from −2 to 6 graphing x > 3: an open circle at 3 and a shaded ray with an arrow to the right x > 3−2−10123456open circle: 3 itself is not a solution

What is an inequality?

An inequality compares two amounts that may not be equal. Like an equation it has two sides, but instead of “=” it uses a symbol such as < or ≥. The solution of an inequality is every number that makes it true, usually infinitely many.

Example 1: Equation or inequality?

Compare x + 2 = 5 and x + 2 > 5. Which values of x work?

  1. x + 2 = 5 is true only when x = 3: one solution.
  2. x + 2 > 5 is true for 4 (4 + 2 = 6 > 5), for 3.5, for 100, for every number greater than 3.
  3. Its solution is x > 3. 3 itself does not work: 3 + 2 = 5, which is not greater than 5.

The symbols and how to read them

  • a < b: a is less than b.
  • a > b: a is greater than b.
  • a ≤ b: a is less than or equal to b.
  • a ≥ b: a is greater than or equal to b.
  • a ≠ b: a is not equal to b.

The open side of < or > faces the bigger number; the point faces the smaller one. Read left to right: 2 < 7 says “2 is less than 7.” The same fact can be flipped around: 7 > 2. In the same way, x > 3 and 3 < x mean the same thing.

Example 2: True or false?

Decide: (a) −4 < −1, (b) 6 ≤ 6, (c) −3 > 2, (d) 5 ≠ 5.

  1. (a) −4 is left of −1 on the number line, so it is smaller: true. (See Integers.)
  2. (b) 6 is equal to 6, and ≤ allows equal: true.
  3. (c) A negative is less than a positive: false.
  4. (d) 5 equals 5: false.

Graphing on a number line

Draw a circle at the boundary number. Use an open circle for < or > (the boundary is not included) and a closed (filled) circle for ≤ or ≥ (it is included). Then shade the side that works, with an arrow to show it goes on forever. If x is written first, the arrow points the same way as the symbol: x < 2 points left.

Four number lines from −2 to 5: x < 2 has an open circle at 2 and points left; x ≤ 2 has a closed circle at 2 and points left; x > 2 has an open circle at 2 and points right; x ≥ 2 has a closed circle at 2 and points right x < 2−2−1012345x ≤ 2−2−1012345x > 2−2−1012345x ≥ 2−2−1012345
Open circles for < and >; closed circles for ≤ and ≥. The arrow shows which side is shaded.

Example 3: Read a graph

A number line has a closed circle at −1 and is shaded to the left. Write the inequality.

  1. Closed circle: −1 is included, so use ≤ or ≥.
  2. Shaded left: numbers smaller than −1.
  3. Answer: x ≤ −1.

Writing inequalities from words

  • “more than,” “greater than,” “above”: >
  • “less than,” “fewer than,” “below”: <
  • “at least,” “no less than,” “minimum”: ≥
  • “at most,” “no more than,” “maximum”: ≤

Example 4: Words to inequalities

Write an inequality: (a) You must be at least 13 to join (age a). (b) A bag can hold no more than 20 kg (weight w). (c) More than 30 people came (number p).

  1. (a) “At least 13” means 13 or older: a ≥ 13.
  2. (b) “No more than 20” means 20 or less: w ≤ 20.
  3. (c) “More than 30” does not include 30: p > 30.

Solving one-step inequalities

Solve inequalities just like equations: use inverse operations to get x alone, doing the same thing to both sides. Adding or subtracting any number, or multiplying or dividing by a positive number, keeps the sign the same. There is one new rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Example 5: One-step inequalities

Solve (a) x − 7 > −2 and (b) 4x ≤ 28.

  1. (a) Add 7 to both sides: x > −2 + 7, so x > 5.
  2. (b) Divide both sides by 4 (positive, so the sign stays): x ≤ 7.

Example 6: Dividing by a negative

Solve −3x < 12.

  1. Divide both sides by −3. It is negative, so < flips to >.
  2. 12 ÷ (−3) = −4, so x > −4.
  3. Check with x = 0 (it should work): −3 × 0 = 0, and 0 < 12 ✓. Check x = −5 (it shouldn’t): −3 × (−5) = 15, and 15 < 12 is false ✓.

How do we know? Why the sign flips

A numeric check: start with something true, 2 < 5. Multiply both sides by −1: you get −2 and −5. But −2 < −5 is false; −2 is to the right of −5. The true statement is −2 > −5. The sign had to flip.

A picture: multiplying by −1 reflects every number across 0, like a mirror. A number that was to the left ends up to the right, so the order of any two numbers reverses. Multiplying by another negative, like −3, is the same mirror followed by a stretch by 3, and stretching doesn’t change the order. Dividing by a negative works the same way, because dividing by −3 is multiplying by −1/3.

A number line from −6 to 6. The points 2 and 5 are multiplied by −1 and land at −2 and −5, mirrored across 0. 2 is left of 5, but −2 is right of −5, so the inequality sign flips 2 < 5, but −2 > −5−6−5−4−3−2−10123456× (−1)× (−1) mirrors the line at 0, so the order reverses
Multiplying by −1 mirrors the number line across 0. The order of every pair reverses.

Two-step inequalities

Undo the addition or subtraction first, then the multiplication or division, just like a two-step equation. Watch for the sign of the coefficient in the last step.

Example 7: Two steps, with a flip

Solve −3x + 4 ≤ 10 and graph it.

  1. Subtract 4 from both sides: −3x ≤ 6.
  2. Divide by −3, which is negative, so ≤ flips to ≥: x ≥ −2.
  3. Graph: a closed circle at −2, shaded to the right. Answer: x ≥ −2.

Checking a solution by testing a value

Pick an easy number from your shaded side and put it into the original inequality: it should make it true. Then try one from the other side: it should make it false. This catches a forgotten flip.

A number line from −5 to 3 graphing x ≥ −2: a closed circle at −2 and a ray to the right. Testing 0 works; testing −4 does not −3x + 4 ≤ 10 → x ≥ −2−5−4−3−2−10123test 0 ✓test −4 ✗closed circle: −2 is a solution
Test points: 0 is in the shaded part and works; −4 is outside and doesn’t.

Inequalities in real life

A ride requires riders to be at least 48 inches tall (h ≥ 48). A speed limit of 30 mph means your speed s ≤ 30. A budget of $40 means what you spend is no more than 40.

A number line from 44 to 52 inches graphing h ≥ 48: a closed circle at 48 and a ray to the right Ride rule: height h ≥ 48 inches444546474849505152“at least 48” includes 48: closed circle
An “at least” rule includes the boundary, so the circle is closed.

Example 8: A budget problem

You have $40. Movie tickets are $9 each and there is a $4 online fee. How many tickets t can you buy?

  1. Total cost is 9t + 4, and it can be at most 40: 9t + 4 ≤ 40.
  2. Subtract 4: 9t ≤ 36. Divide by 9: t ≤ 4.
  3. You can’t buy part of a ticket, so you can buy 0, 1, 2, 3 or 4 tickets. Check 4: 9 × 4 + 4 = 40 ≤ 40 ✓; 5 would cost $49.

History: where < and > come from

The earliest known use of the symbols < and > is in Artis Analyticae Praxis, a book of Thomas Harriot’s algebra printed in 1631, ten years after he died. Many books say Harriot invented them, but MacTutor says the symbols were introduced by the book’s editor, not Harriot himself.

Signs for “less than or equal to” came later. Wikipedia reports that John Wallis used a related mark in 1670 and that the double-barred ≦ and ≧ appeared in the work of the French scientist Pierre Bouguer in 1734. The single-bar ≤ and ≥ used today are a later simplification.

Practice problems

Try each one, then open it to compare.

1. Write with a symbol: “y is at most −2.”

y ≤ −2.

2. Is x = −6 a solution of x > −5?

No. −6 is to the left of −5, so −6 < −5.

3. Graph x < 4: what kind of circle, and which way?

An open circle at 4, shaded to the left.

4. Solve x + 9 ≥ 3.

Subtract 9: x ≥ −6.

5. Solve x/2 > −5.

Multiply by 2 (positive): x > −10.

6. Solve −5x ≥ 35.

Divide by −5 and flip: x ≤ −7. Check x = −8: −5 × (−8) = 40 ≥ 35 ✓.

7. Solve 2x − 7 < 9.

Add 7: 2x < 16. Divide by 2: x < 8.

8. A taxi charges $3 plus $2 per mile. You have $15. Write and solve an inequality for the miles m.

3 + 2m ≤ 15, so 2m ≤ 12 and m ≤ 6: at most 6 miles.

Game: Boundary Hunt

Ten inequalities to solve. Type the answer with x first or last, such as x > 3, 3 < x or x >= -2 (you can type ≥ or >=). Press Enter. Watch out for negatives! One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • Why does an inequality usually have infinitely many solutions, while x + 2 = 5 has one?
  • “No more than 20” and “less than 20”: what is different about them? Draw both.
  • Your friend solved −2x > 8 and got x > −4. How could testing a value show the mistake?
  • Where in your life do you meet a limit that could be written with ≤ or ≥?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Names and prices in the examples are made up for practice.