5-1 Solving One-Step Inequalities

Property

An inequality is used in algebra to compare two quantities that may have different values or to show a range of possible values. We use these four specific symbols for inequalities:

• $<$ is read "$a$ is less than $b$"
• $>$ is read "$a$ is greater than $b$"
• $\leq$ is read "$a$ is less than or equal to $b$"
• $\geq$ is read "$a$ is greater than or equal to $b$"

Examples

• The statement "15 is greater than 9" is written in algebra as $15 > 9$.
• To show that your age, $a$, must be at least 18 to vote, you would write $a \geq 18$.
• The inequality $x < 5$ means that the value of $x$ can be any number that is strictly less than 5.

Explanation

These symbols are used to show the relationship between two values when they are not perfectly equal. They tell us whether one value is bigger or smaller than another, or if there is a limit. A helpful trick is to remember that the small, pointed end of $<$ or $>$ always faces the smaller number, like an arrow pointing to the lesser value. The line underneath $\leq$ and $\geq$ is simply half of an equal sign!

Property

When graphing inequalities on a number line, we use different symbols to show whether the boundary number is included in the solution set.

• For strictly "greater than" ($>$) or "less than" ($<$), use an open circle to show the number is NOT included.
• For "greater than or equal to" ($\geq$) or "less than or equal to" ($\leq$), use a closed circle to show the number IS included.

Then, shade the number line or draw an arrow in the direction of all the numbers that make the inequality true.

Examples

• The inequality $x < 4$ includes all numbers to the left of 4, but not 4 itself. On a number line, we place an open circle at 4 and shade to the left.
• The inequality $y \geq -2$ includes -2 and all numbers greater than it. On a number line, we place a closed (filled) circle at -2 and shade to the right.
• The inequality $p > -0.5$ includes all numbers to the right of -0.5, but not -0.5 itself. On a number line, we place an open circle at -0.5 and shade to the right.

Explanation

Unlike simple equations that usually have just one answer (like $x = 5$), inequalities often have infinitely many solutions! Because we cannot write down an infinite list of numbers, we graph them on a number line to visually represent all possible values at once. The open or closed circle tells us exactly where the solution starts, and the shaded arrow tells us which direction it goes forever.

Property

Solving one-step inequalities is almost identical to solving one-step equations: you use inverse operations to isolate the variable.

• Addition/Subtraction: You can add or subtract the same number from both sides. The inequality sign does not change.
• Multiplication/Division by a Positive: You can multiply or divide both sides by a positive number. The inequality sign does not change.
• Multiplication/Division by a Negative: If you multiply or divide both sides by a negative number, you MUST reverse the inequality symbol ($<$ becomes $>$, $\geq$ becomes $\leq$).

Examples

• Using Subtraction: Solve $x + 5 \geq 12$.

Subtract 5 from both sides: $x \geq 7$. (Sign stays the same)

• Dividing by a Positive: Solve $3x > 15$.

Divide both sides by 3: $x > 5$. (Sign stays the same)

• Dividing by a Negative: Solve $-2y \leq 8$.

Divide both sides by -2. Because you divided by a negative, reverse the sign: $y \geq -4$.

Explanation

For the most part, you can treat an inequality symbol just like an equal sign when trying to get a variable by itself. The only major trap is the "Negative Rule." Why must we flip the sign? Think about the number line: $4 > 2$ is true. But if we multiply both sides by -1, we get $-4$ and $-2$. Since -4 is further to the left on the number line, it is now smaller, so we must flip the sign to make the statement true: $-4 < -2$.