Solving Linear Inequalities in One Variable

Solving by isolating the variable

You can picture a solution set on a number line. A closed circle means the endpoint is included ($\le$ or $\ge$, written with a bracket). An open circle means it is not included ($<$ or $>$, written with a parenthesis). Infinity always gets a parenthesis.

To test whether a number belongs to the solution set, substitute it into the original inequality and see if the statement is true. For $4x - 7 \ge 2x + 1$, try $x = 5$: $13 \ge 11$ is true, so 5 is in the solution set. Try $x = 0$: $-7 \ge 1$ is false, so 0 is not.

Parentheses, fractions, and decimals

Clean the inequality up first. Distribute to remove parentheses, then combine like terms on each side. If there are fractions, multiply every term by the least common denominator (LCD). The LCD is positive, so the sign does not flip in that step. For decimals, multiply by 10, 100, and so on, to make them whole numbers.

When the variable disappears

Sometimes the $x$ terms cancel. Then you are left with a statement made only of numbers, and it is either always true or always false.

Each step in these solutions turns the inequality into an equivalent one (same solutions). That is exactly how you justify an answer analytically: list the steps and say why each is allowed.

Translating words into inequalities

Most real problems give you a sentence, not a symbol. Learn the key phrases:

"Between 48 and 76 inclusive" means two conditions at once: $48 \le x \le 76$.

Follow four steps: (1) define the variable, (2) translate the sentence into an inequality, (3) solve it, (4) interpret the answer in context (units, whole numbers, impossible values).

Comparing plans, averages, and mixtures

Three more situations show up again and again. For two plans, write each cost as an expression in the same variable, then ask which is smaller. For an average, remember that the average of $n$ numbers is their sum divided by $n$. For mixtures and interest, the amount of the substance is percent × total.

Here is the ride-share comparison below as a graph. Where one line is below the other, that plan is cheaper.

When a plan charges for extra units (for example, extra gigabytes), the extra cost is rate × (amount used − amount included), and it only applies when the amount used is above the included amount.

Interest problems work the same way. If you split $\$5{,}000$ between a 3% account and a 5% account, and $x$ goes into the 5% account, then $5000 - x$ is in the 3% account. The total interest after one year is $0.05x + 0.03(5000 - x)$.

Back to Emma: she needs $45 + 12h \ge 120$. Subtract 45: $12h \ge 75$. Divide by 12: $h \ge 6.25$. She has to work at least 6.25 hours, which is 6 hours 15 minutes.

Inequalities with a parameter

Sometimes an inequality has another letter besides $x$, called a parameter (like $k$ or $m$). After you collect the $x$ terms, the coefficient of $x$ may involve the parameter. Its sign decides whether you flip, so you must treat each case separately.

Finding a parameter and literal inequalities

Sometimes the problem goes backward. You are told what the solution set is, and you must find the parameter. The plan: solve for $x$ in terms of the parameter, set the endpoint equal to the given number, solve for the parameter, and check.

A literal inequality uses letters for all the numbers. You solve it the same way, as long as the conditions you are given tell you the signs.

Summary