Solve and Graph One-Step Inequalities | Grade 7 Math | LevelKite

An inequality compares two amounts using $<$ , $>$ , $\le$ or $\ge$ . In this lesson, $x$ stands for an unknown number, and it can be any number, including negatives and fractions. A solution is a value of $x$ that makes the inequality true. For $x + 4 > 9$ , the value $6$ is a solution, because $6 + 4 = 10$ and $10 > 9$ . The value $2$ is not, because $2 + 4 = 6$ and $6 > 9$ is false. The solution set is all of the solutions together.

You already know how to graph an inequality with $x$ alone on one side. In a one-step inequality such as $x + 4 > 9$ , one operation is done to $x$ . To solve it, get $x$ alone by doing the inverse operation, the operation that undoes the one done to $x$ , on both sides. The question is whether the inequality symbol stays the same or reverses, meaning it points the other way, as when $<$ becomes $>$ . Try it on the true statement $2 < 6$ .

Start with the true statement 2 < 6 and do the same thing to both sides.

Do this to both sides	New left side (from 2)	New right side (from 6)	True result	Symbol
Add $3$	$2 + 3 = 5$	$6 + 3 = 9$	$5 < 9$	stays
Subtract $3$	$2 - 3 = -1$	$6 - 3 = 3$	$-1 < 3$	stays
Multiply by $2$	$2 \cdot 2 = 4$	$6 \cdot 2 = 12$	$4 < 12$	stays
Divide by $2$	$2 \div 2 = 1$	$6 \div 2 = 3$	$1 < 3$	stays
Multiply by $-1$	$2 \cdot (-1) = -2$	$6 \cdot (-1) = -6$	$-2 > -6$	reverses
Divide by $-2$	$2 \div (-2) = -1$	$6 \div (-2) = -3$	$-1 > -3$	reverses

Picture each pair of numbers on a number line, where the smaller number is the one farther left.

Adding or subtracting the same number slides both numbers the same distance in the same direction. The left one stays on the left, so the symbol stays.
Multiplying or dividing by a positive number zooms the number line in or out around $0$ . The numbers spread apart or squeeze together, but they keep their order, so the symbol stays.
Multiplying or dividing by a negative number gives the opposite of the result for the matching positive number. For example, $6 \div (-2) = -3$ is the opposite of $6 \div 2 = 3$ . Taking opposites flips the number line across $0$ like a mirror, which turns the order around: $2$ is left of $6$ , but $-2$ is right of $-6$ . So the symbol must reverse to keep the statement true.

Key concept

Keep or reverse the symbol, then test

Solve with one inverse operation on both sides.

Keep the symbol when you add or subtract any number, or multiply or divide by a positive number.
Reverse it when you multiply or divide by a negative number.

After you solve, the number across the symbol from $x$ is the boundary. The symbols $\le$ and $\ge$ include it, so testing it gives true (filled circle); $<$ and $>$ exclude it, so the test gives false (hollow circle). Test a value on each side: the ray of solutions runs toward the side whose value makes the inequality true. Then check a value on the ray in the original inequality. When $x$ can be any number, the ray shows the whole solution set; a real-world context may allow only whole numbers or only nonnegative numbers (zero or greater).

Worked example

Solve and graph $x - 3 \ge -1$ , where $x$ can be any number.

Undo subtracting 3 by adding 3 to both sides. Adding the same number slides both sides the same distance, so their order is kept and ≥ stays.
$x - 3 + 3 \ge -1 + 3$
Simplify each side. On the left, −3 + 3 = 0, which leaves x. On the right, −1 + 3 = 2. The boundary is 2.
$x \ge 2$
Ask whether the boundary 2 is a solution. It is, because ≥ includes equal values, so the circle at 2 is filled.
$2 \ge 2 \ \text{(true)}$
Test a value on each side of 2. The value 0 makes the solved inequality false and 4 makes it true, so the ray runs right from 2.
$0 \ge 2 \ \text{(false)}, \qquad 4 \ge 2 \ \text{(true)}$
Check 4, a value on the ray, in the original inequality. It makes the original true, as every value on the ray should.
$4 - 3 = 1, \qquad 1 \ge -1 \ \text{(true)}$

Answer

The solutions are all numbers greater than or equal to $2$ , so $x \ge 2$ , graphed as a filled circle at $2$ with a ray running right.

The filled circle at

2

and the ray running right show

x \ge 2

.

Worked example

Solve and graph $3x < 7$ , where $x$ can be any number.

3x means 3 times x, so undo it by dividing both sides by 3. Dividing by a positive number keeps the order of the two sides, so < stays. On the left, 3x ÷ 3 = x.
$\frac{3x}{3} < \frac{7}{3}, \quad \text{so} \quad x < \frac{7}{3}$
Locate the boundary. Since 7 ÷ 3 = 2 remainder 1, the boundary is two and one third, which is one third of the way from 2 to 3. It is not a whole number, so a number line marked at whole numbers has no tick for it. Describe its place in words instead.
$\frac{7}{3} = 2\tfrac{1}{3}$
The boundary is not a solution, because < excludes equal values, so the circle is hollow. On each side, 2 makes the solved inequality true and 3 makes it false, so the ray runs left.
$\tfrac{7}{3} < \tfrac{7}{3} \ \text{(false)}; \quad 2 < 2\tfrac{1}{3} \ \text{(true)}; \quad 3 < 2\tfrac{1}{3} \ \text{(false)}$
Check 2, a value on the ray, in the original inequality. It makes the original true.
$3 \cdot 2 = 6, \qquad 6 < 7 \ \text{(true)}$

Answer

The solutions are all numbers less than $\frac{7}{3}$ , so $x < \frac{7}{3}$ . In words, the graph is a hollow circle one third of the way from $2$ to $3$ , with a ray running left.

Worked example

Solve and graph $-2x < 6$ , where $x$ can be any number.

−2x means −2 times x, so divide both sides by −2. Dividing by a negative number flips the order of the two sides, so < reverses to >.
$\frac{-2x}{-2} > \frac{6}{-2}$
Simplify. On the left, −2x ÷ (−2) = x. On the right, a positive number divided by a negative number is negative, so 6 ÷ (−2) = −3.
$x > -3$
The boundary −3 is not a solution, because > excludes equal values, so the circle is hollow. On each side, −4 makes the solved inequality false and 0 makes it true, so the ray runs right.
$-3 > -3 \ \text{(false)}; \quad -4 > -3 \ \text{(false)}; \quad 0 > -3 \ \text{(true)}$
Check −2, a value on the ray, in the original inequality. It makes the original true.
$-2 \cdot (-2) = 4, \qquad 4 < 6 \ \text{(true)}$

Answer

The solutions are all numbers greater than $-3$ , so $x > -3$ , graphed as a hollow circle at $-3$ with a ray running right.

The hollow circle at

-3

and the ray running right show

x > -3

.

Takeaway: Solve a one-step inequality with one inverse operation on both sides. The symbol stays when you add, subtract, or multiply or divide by a positive number; it reverses when you multiply or divide by a negative number. Then let tests build the graph: $\le$ and $\ge$ include the boundary, $<$ and $>$ exclude it, the ray runs toward the side that tests true, and a value on the ray should make the original true. Next, you will solve one-step inequality word problems, where the context may allow only whole or nonnegative solutions.