Linear Inequalities

What is an inequality?

An equation says two things are equal. An inequality compares two things that might not be equal: one side is smaller or bigger than the other. There are four inequality symbols.

To remember which is which: the open, wide side of < or > always faces the bigger number. In 3 < 8, the wide side faces 8.

The line under ≤ and ≥ is half of an equals sign. It means “or equal to,” so x ≤ 10 allows 10 itself, but x < 10 doesn’t.

A linear inequality is like a linear equation, but with one of these four symbols instead of =. The variable only appears to the first power: 3x − 5 ≥ 7 is linear, but x² < 9 is not.

Solutions are sets

A solution of an inequality is a number that makes it true. Most linear equations have exactly one solution. Most inequalities have infinitely many.

Take x > 3. The numbers 4, 10, 3.5, 3.01 and a million are all solutions. 3 is not, because 3 is not greater than itself. The collection of every solution is called the solution set.

Here’s how to test some numbers in 2x − 1 < 5: put each one in for x and see whether the statement comes out true.

Solving gives the whole set at once: 2x − 1 < 5 becomes x < 3. That matches the table. −2 and 0 are below 3, while 3 and 5 are not.

Later courses sometimes write a solution set in set-builder notation: {x | x < 3} is read “the set of all x such that x is less than 3.” On this page we’ll mostly use plain inequalities, number-line graphs and interval notation.

You can’t list infinitely many solutions, but you can draw them. A number-line graph uses two marks:

• A circle at the boundary number. It’s open (hollow) for < and >, because the boundary is not a solution. It’s closed (filled in) for ≤ and ≥, because the boundary is a solution.
• Shading over every solution. x > 3 and x ≥ 3 shade to the right, toward bigger numbers. x < 3 and x ≤ 3 shade to the left. The arrow means “and so on, forever.”

Try it yourself. Choose a type of inequality, a symbol and a number, and the grapher draws it. Then test numbers to see which ones are solutions. You can also tap the number line to place the test point.

You solve an inequality with the same moves as an equation: add, subtract, multiply or divide both sides by the same number. The goal is still to get x alone. Adding or subtracting never changes the direction of the sign, and neither does multiplying or dividing by a positive number.

Your turn

Here is the one place where inequalities behave differently from equations. Start with a true statement, 2 < 5, and do the same thing to both sides. Watch what happens to the sign.

Multiplying by −1 turns 2 into −2 and 5 into −5. But −2 is greater than −5: on a number line, −2 is to the right. If you kept the old sign, you’d get −2 < −5, which is false.

Multiply or divide by a negative → flip the sign.

So < becomes >, > becomes <, ≤ becomes ≥, and ≥ becomes ≤.

Your turn

Use the same order as with equations: distribute and combine like terms first, undo adding or subtracting next, then multiply or divide. Only flip at the step where you multiply or divide by a negative.

Your turn

Collect the x-terms on one side and the numbers on the other. Here’s a handy trick: move the smaller x-term. Then the x-term that’s left has a positive coefficient, and you won’t need to flip at all.

You could also do Example 2 by subtracting 6x instead. You’d get −4x + 9 ≤ −3, then −4x ≤ −12, and dividing by −4 (with a flip) gives the same answer, x ≥ 3. Both ways are correct. The trick just saves you a flip.

Your turn

Sometimes the x-terms cancel and you’re left with a statement with no x at all. Then the answer depends only on whether that statement is true.

The graph of “all real numbers” is the whole line, shaded, with arrows at both ends. For “no solution” nothing is shaded.

A compound inequality joins two inequalities with and or or.

• And means both must be true. “x is greater than −2 and at most 3” describes the numbers between −2 and 3. We write it in one chain: −2 < x ≤ 3. Read it as “x is between −2 and 3, including 3.”
• Or means at least one must be true. x < −1 or x ≥ 4 describes numbers outside the stretch from −1 to 4.

To solve a chain like −3 < 2x + 1 ≤ 7, do the same thing to all three parts at once, so that x is alone in the middle. If you divide by a negative, flip both signs.

Your turn

Interval notation

Interval notation is a short way to write a solution set, used a lot in later courses. You write the left end and the right end of the shaded part, separated by a comma.

• A parenthesis ( or ) means that end is not included, like an open circle.
• A bracket [ or ] means that end is included, like a closed circle.
• ∞ (infinity) and −∞ mean the shading goes on forever. They always get a parenthesis, because infinity isn’t a number you can reach.
• Write the smaller end first. Use ∪ (“union”) to join the two pieces of an “or.”

Checking with test points

With an equation, you check one number. With an inequality, check three things:

1. The boundary. Put the boundary number in. The two sides should come out equal.
2. A test point inside the shaded part. It should make the original inequality true.
3. A test point outside the shaded part. It should make the original inequality false.

Suppose you solved 4 − 3x ≤ 10 and got x ≥ −2. Here’s the check, always in the original inequality:

Test points are great at catching a forgotten flip. If you had written x ≤ −2 instead, −5 would be inside your answer, but the table shows −5 doesn’t work. That tells you the sign points the wrong way.

Word problems use phrases instead of symbols. These are the ones you’ll meet most:

Careful: “5 less than a number” is the expression n − 5, not an inequality. It’s “is less than” that means <.

The steps are the same as for equation word problems: name the variable, write the inequality, solve it, then answer the question in a sentence. Make sure the answer makes sense, since you can’t buy half a ticket or send −3 texts.

Your turn

Common mistakes

Solve each one, then open the answer to check your steps and the graph. Numbers 1–4 are one step, 5–9 are multi-step, 10–11 are special cases, 12–13 are compound, and 14 is a word problem.

History: where do < and > come from?

The signs < and > first appeared in print in 1631, in an algebra book by the English mathematician Thomas Harriot (about 1560–1621). Its Latin title is Artis Analyticae Praxis, roughly “The Practice of the Analytic Art.” The book explains the new sign with the example a > b, meaning “a is greater than b.”

Harriot never saw it. He died in 1621, and the book was put together from his papers ten years later, mainly by his friend Walter Warner. Historians who have studied Harriot’s handwritten pages say he used different-looking signs himself. So the shapes we use today may have been chosen by the editors, not by Harriot.

Harriot did much more than algebra. He taught navigation to the sailors on Sir Walter Raleigh’s voyages, sailed to Roanoke Island in what’s now North Carolina in 1585, and on 26 July 1609 sketched the Moon through a telescope, the earliest telescope observation known in England.

Sources

• Jeff Miller, Earliest Uses of Symbols of Relation, hosted by MacTutor History of Mathematics, University of St Andrews (Harriot’s < and > in 1631, with the Latin explanation; Wallis, 1670; Bouguer, 1734; the manuscript question).
• J. J. O’Connor and E. F. Robertson, MacTutor: Thomas Harriot biography (life and dates, the 1585 voyage, the 1609 Moon sketch, the posthumous 1631 book and its editors).
• Muriel Seltman and Robert Goulding, Thomas Harriot’s Artis Analyticae Praxis: An English Translation with Commentary (Springer, 2007; publisher’s book page), whose introduction and commentary compare the 1631 book with Harriot’s surviving manuscript papers.
• Common Core State Standards for Mathematics: Reasoning with Equations and Inequalities (A-REI.B.3, solving linear inequalities in one variable) and Creating Equations (A-CED.A.1, writing inequalities for word problems), which this lesson supports.

Credits: all illustrations and number-line graphs on this page are original SVG drawings made for Jugami. Every worked example and answer was checked by testing numbers in the original inequality and in every step.