Solve and Graph Linear Inequalities | Geometry | LevelKite

You already know how to solve two-step equations, and that skill will carry you most of the way here. An equation says that two expressions are equal. An inequality says how two expressions compare, using $<$ (is less than), $>$ (is greater than), $\le$ (is less than or equal to) or $\ge$ (is greater than or equal to).

A solution of an inequality is a value of the variable that makes it true. For $x + 3 < 7$ , the value $1$ is a solution, because $1 + 3 = 4$ and $4 < 7$ is true. The value $4$ is not a solution, because $4 + 3 = 7$ and $7 < 7$ is false. This inequality has infinitely many solutions, including $3.5$ and $-10$ , so you cannot simply list them. The solution set is the set of all solutions, and to solve an inequality means to find that set.

Why do the solving moves work? On a number line, $a < b$ means that $a$ lies to the left of $b$ . Adding or subtracting the same number slides both points the same distance in the same direction, so their order stays.

Multiplying by a positive number moves points away from or toward $0$ without swapping them. From $2 < 5$ , multiplying by $3$ gives $6 < 15$ . Multiplying by a negative number also sends each nonzero point to the opposite side of $0$ , like a mirror image, and a mirror swaps left and right. From $2 < 5$ , multiplying by $-3$ gives $-6$ and $-15$ , and $-6 > -15$ .

Dividing by a nonzero number $c$ means multiplying by $\frac{1}{c}$ , which has the same sign as $c$ , so division follows the same pattern.

Key concept

When the symbol stays and when it reverses

Solving replaces an inequality with simpler equivalent inequalities, which have exactly the same solution set. You use each property below in its if-then direction, and each move can be undone, so the solutions do not change.

Addition and Subtraction Properties of Inequality: adding or subtracting the same number on both sides keeps the symbol.
Multiplication and Division Properties of Inequality: multiplying or dividing both sides by a positive number keeps the symbol. A negative number reverses it: $<$ and $>$ trade places, and $\le$ and $\ge$ trade places.
Zero is excluded: division by zero is undefined, and multiplying both sides by $0$ erases the variable.

The formula shows $<$ ; the other symbols follow the same pattern.

\text{If } a < b \text{ and } c > 0, \text{ then } ac < bc. \quad \text{If } a < b \text{ and } c < 0, \text{ then } ac > bc.

A one-step inequality needs one move to reach solved form, with the variable alone on one side. Here are three quick cases:

For $x - 7 > -2$ , add $7$ to both sides. The Addition Property of Inequality keeps the symbol, so $x > 5$ .
For $3x < -12$ , divide both sides by $3$ . This divisor is nonzero and positive, so the Division Property of Inequality keeps the symbol, and $x < -4$ . The negative number on the right does not change that.
For $\frac{x}{-2} \ge 3$ , multiply both sides by $-2$ . This number is negative, so the Multiplication Property of Inequality reverses the symbol, and $x \le -6$ .

Check each result in the original inequality. The number in solved form is the boundary. It should make both sides equal, and a value from your solution set should make the inequality true.

Worked example

A one-step inequality with a negative divisor

Given $-4x \le 20$ , find the solution set.

The variable is multiplied by −4, so divide both sides by −4. This divisor is nonzero, so the division is defined. It is negative, so the Division Property of Inequality reverses ≤ to ≥.
$\frac{-4x}{-4} \ge \frac{20}{-4}$
Simplify each side. A nonzero number divided by itself is 1, so −4x ÷ (−4) = x, and 20 ÷ (−4) = −5.
$x \ge -5$
Check in the original inequality. The boundary −5 makes both sides equal to 20. The value 0 is in your solution set, and it makes the inequality true.
$-4(-5) = 20 \qquad -4(0) = 0,\ \text{and } 0 \le 20$

Answer

The solution set is all real numbers greater than or equal to $-5$ , written $x \ge -5$ .

Worked example

A two-step inequality

Given $5 - 2x > 11$ , find the solution set.

This inequality needs two moves. Undo the 5 first, as you would in a two-step equation: subtract 5 from both sides. The Subtraction Property of Inequality keeps the symbol.
$5 - 2x - 5 > 11 - 5 \quad\Rightarrow\quad -2x > 6$
Divide both sides by −2, which is nonzero. Because −2 is negative, the Division Property of Inequality reverses > to <.
$\frac{-2x}{-2} < \frac{6}{-2}$
Simplify each side: −2x ÷ (−2) = x, and 6 ÷ (−2) = −3.
$x < -3$
Check in the original inequality. The boundary −3 makes both sides equal to 11. The value −4 is in your solution set, and it makes the inequality true.
$5 - 2(-3) = 11 \qquad 5 - 2(-4) = 13,\ \text{and } 13 > 11$

Answer

The solution set is all real numbers less than $-3$ , written $x < -3$ .

A multi-step inequality takes more than two moves to reach solved form. Use a dependable order. First, simplify each side with the Distributive Property and by combining like terms, which have the same variable part or are both plain numbers. Next, use the Addition and Subtraction Properties of Inequality to gather the variable terms on one side and the numbers on the other. Last, divide both sides by the variable's coefficient, the number multiplying it. The coefficient must be nonzero, as it is in every inequality in this lesson. A positive coefficient keeps the symbol, and a negative coefficient reverses it.

Worked example

A multi-step inequality

Given $3(x - 2) + 4 \le 5x + 6$ , find the solution set.

Remove the parentheses with the Distributive Property: 3(x − 2) = 3x − 6.
$3x - 6 + 4 \le 5x + 6$
Combine the like terms on the left: −6 + 4 = −2. The value of that side does not change.
$3x - 2 \le 5x + 6$
Subtract 5x from both sides to gather the variable terms on the left. For any value of x, 5x is a number, so the Subtraction Property of Inequality applies and keeps the symbol.
$-2x - 2 \le 6$
Add 2 to both sides. The Addition Property of Inequality keeps the symbol.
$-2x \le 8$
Divide both sides by −2, which is nonzero. Because −2 is negative, the Division Property of Inequality reverses ≤ to ≥.
$\frac{-2x}{-2} \ge \frac{8}{-2} \quad\Rightarrow\quad x \ge -4$
Check in the original inequality. The boundary −4 makes both sides equal to −14. The value 0 is in your solution set, and it makes the inequality true.
$3(-4-2)+4 = -14 = 5(-4)+6 \qquad 3(0-2)+4 = -2 \le 6 = 5(0)+6$

Answer

The solution set is all real numbers greater than or equal to $-4$ , written $x \ge -4$ .

To graph a solution set, you mark on a number line exactly the points that are solutions. Start from solved form with the variable on the left. If your work ends as $-4 \le x$ , rewrite it as $x \ge -4$ , because $a \le b$ and $b \ge a$ say the same thing.

The graph is then a thick, shaded ray: a path that starts at the boundary and runs forever in one direction, as its arrowhead shows. A circle at the boundary tells you whether that starting point belongs to the set. A strict inequality uses $<$ or $>$ , so the boundary is not a solution and you draw an open circle, which is hollow. Otherwise the symbol is $\le$ or $\ge$ , so the boundary is a solution and you draw a closed circle, which is filled. The shaded ray must agree with the solved inequality: greater values lie to the right of the boundary, and lesser values lie to the left.

The four solved forms, where a is the boundary

Solved form	Strict?	Circle at the boundary	Ray runs
$x < a$	Yes	Open	Left
$x \le a$	No	Closed	Left
$x > a$	Yes	Open	Right
$x \ge a$	No	Closed	Right

Here is the graph of $x \ge -4$ , the result of the multi-step example. The boundary is $-4$ . The symbol $\ge$ is not strict, and your check showed that $x = -4$ makes both sides of the original inequality equal, so $-4$ is a solution and its circle is closed. Numbers greater than $-4$ lie to its right, so the ray runs right. A graph records what the algebra showed, so decide the circle and the ray from the solved inequality, never from how a drawing looks.

The closed (filled) circle on

-4

and the ray to the right show

x \ge -4

.

Next is the graph of $x < -3$ , the result of the two-step example. The boundary is $-3$ . The symbol $<$ is strict: at $x = -3$ , both sides of $5 - 2x > 11$ equal $11$ , and $11 > 11$ is false. So $-3$ is not a solution, and its circle is open. Numbers less than $-3$ lie to its left, so the ray runs left.

You can test this graph in the original inequality. The value $-5$ is on the ray, and $5 - 2(-5) = 15$ , with $15 > 11$ true. The value $-2$ is off the ray, and $5 - 2(-2) = 9$ , with $9 > 11$ false.

The open (hollow) circle on

-3

and the ray to the left show

x < -3

.

A constraint is a condition that a quantity must meet. To write an inequality from a constraint, name a variable and say what it measures, then build each side from the stated facts. The words choose the symbol:

at most or no more than means $\le$
at least or no less than means $\ge$
is less than means $<$
is more than means $>$

After you solve, check the result against the geometry too. A side length must be positive, and each angle of a triangle must measure between $0^\circ$ and $180^\circ$ , even when the inequality alone allows other values.

Worked example

Designing a fenced garden

You are designing a rectangular garden with fencing all the way around it. Given: a 50-foot roll of fencing, a garden length of 12 feet, and a requirement that at least 8 feet of fencing be left over. Find: every width $w$ , in feet, that meets these constraints.

Let w be the width in feet. By the definition of perimeter, the fencing used is the sum of the four side lengths. Opposite sides of a rectangle are equal, so the sides are 12, 12, w and w.
$12 + 12 + w + w = 24 + 2w$
Leftover fencing is the roll minus the fencing used. The words at least 8 mean greater than or equal to 8.
$50 - (24 + 2w) \ge 8$
Apply the Distributive Property to the subtraction, then combine the numbers on the left: 50 − 24 = 26.
$50 - 24 - 2w \ge 8 \quad\Rightarrow\quad 26 - 2w \ge 8$
Subtract 26 from both sides. The Subtraction Property of Inequality keeps the symbol.
$-2w \ge -18$
Divide both sides by −2, which is nonzero. Because −2 is negative, the Division Property of Inequality reverses ≥ to ≤.
$\frac{-2w}{-2} \le \frac{-18}{-2} \quad\Rightarrow\quad w \le 9$
Check the result. At the boundary w = 9, exactly 8 feet are left, which at least 8 allows. At w = 5, 16 feet are left, and 16 ≥ 8. The inequality alone allows negative numbers, but a width must be positive, so w > 0 as well.
$50-(24+2\cdot 9)=8 \qquad 50-(24+2\cdot 5)=16$

Answer

The garden meets the constraints when its width is greater than $0$ feet and at most $9$ feet.

Takeaway: An inequality has a whole set of solutions, and each solving move must keep that set the same. Keep the symbol when you multiply or divide both sides by a positive number, reverse it for a negative number, and never divide by zero. Then graph the result with an open circle for a strict inequality and a closed circle otherwise, with a ray that agrees with your solved inequality.