linear inequalities

Linear inequalities unlock real-world constraints—on the SAT, they ask you about solution sets, regions, and “which values work?” questions. This page goes deep: learn how to solve and graph, handle reversals with negatives, model with word problems, and conquer SAT-level traps.

what is a linear inequality?

A linear inequality is just like a linear equation, but instead of an equals sign, it uses a symbol like $<$ , $>$ , $\leq$ , or $\geq$ . The solution to a linear inequality isn’t just one number, but a whole range or interval of values.

Example:

$3x + 5 < 11$ asks for all values of $x$ that make the inequality true.

3x + 5 < 11 \\ 3x < 6 \\ x < 2

So every $x$ less than 2 is a solution. On a number line, this is all points to the left of 2.

Linear inequalities always describe a region of possible answers, not just one value—think intervals, not points!

solving & graphing: one variable

To solve a linear inequality, use your usual algebra skills: combine like terms, isolate $x$ , and simplify. But watch out—if you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example:

-2x + 6 \geq 10 \\ -2x \geq 4 \\ x \leq -2

When you divide both sides by $-2$ , the inequality flips from $\geq$ to $\leq$ . The solution is all $x$ less than or equal to $-2$ .

On a number line, you’d draw a closed circle at $-2$ and shade everything to the left for $x \leq -2$ . Use an open circle for $<$ or $>$ and a closed circle for $\leq$ or $\geq$ .

Always reverse the sign if you multiply or divide both sides by a negative!

compound & absolute value inequalities

Compound inequalities are created when two inequalities are joined by "and" (meaning both conditions must be true—an intersection), or "or" (meaning either condition can be true—a union). For example, the solution to $2 < x \leq 7$ is all values of $x$ that are greater than 2 and less than or equal to 7.

Absolute value inequalities require you to "break up" the absolute value. For instance, $|ax + b| < c$ means $-c < ax + b < c$ . You solve both sides to find the interval.

Example:

|2x-5| < 7 \\ -7 < 2x-5 < 7 \\ -2 < 2x < 12 \\ -1 < x < 6

Here, we split the absolute value into two inequalities: $-7 < 2x - 5 < 7$ . Then solve each part to get the final answer: $-1 < x < 6$ .

Always remember: if $|A| < B$ , then $-B < A < B$ . For "greater than," you split into two separate inequalities: $A > B$ or $A < -B$ .

linear inequalities in two variables

A linear inequality in two variables, like $y > 2x - 1$ , doesn’t just have a single answer or a line as its solution. Instead, the solution is an entire region of the coordinate plane. To graph an inequality, first draw the boundary line $y = 2x - 1$ . Use a dashed line for " $<$ " or " $>$ " (since the boundary isn’t included), and a solid line for " $\leq$ " or " $\geq$ " (since the boundary is included).

Next, decide which side of the line to shade. For inequalities like " $y >$ " or " $y \geq$ ", you shade above the line. For " $y <$ " or " $y \leq$ ", you shade below.

Example:

y > 2x - 1

Draw a dashed line for $y = 2x - 1$ . To check which side to shade, use a test point—usually the origin, $(0,0)$ .

0 > 2 \cdot 0 - 1

Is $0 > -1$ ? Yes, so the origin is part of the solution region. Shade the side of the line that includes $(0,0)$ .

The solution is every point above (but not on) the line $y = 2x - 1$ .

systems of linear inequalities

A system of linear inequalities involves two or more inequalities in two variables, like $x$ and $y$ . Instead of a single line, the solution is a region of the plane—specifically, all the points that satisfy every inequality in the system at the same time. SAT problems often use these to model real-life limits, such as budgets, minimums, or maximums.

Example:

For a charity event, a school sells child tickets for $8 each and adult tickets for $12 each. The school must sell at least 15 child tickets ( $x$ ) and at least 10 adult tickets ( $y$ ). The auditorium can seat no more than 60 people, and the total ticket revenue must be at least $600. Write a system of inequalities to model this situation.

Child tickets: $x \geq 15$
Adult tickets: $y \geq 10$
Total seats: $x + y \leq 60$
Total revenue: $8x + 12y \geq 600$

\begin{cases} x \geq 15 \\ y \geq 10 \\ x + y \leq 60 \\ 8x + 12y \geq 600 \end{cases}

The solution is every combination of child and adult tickets (where both values are whole numbers) that meets all four conditions: enough of each ticket, not over capacity, and hitting the revenue goal.

On the SAT, the solution to a system is the overlapping region on the graph—or, in real-world problems, every combination that fits all requirements!

word problems with linear inequalities

SAT word problems involving linear inequalities ask you to translate real-life limits or requirements into math. Usually, you’ll see phrases like “at least,” “no more than,” or “must be greater than.” These can involve one or two variables, and the solution shows all the values that fit the scenario.

Always define your variables, write an inequality for the constraint, and answer what the question is really asking.

Example:

Sarah has at most $50 to spend on movie tickets. If each ticket costs $8, what is the greatest number of tickets she can buy?

Let $x$ be the number of tickets. The cost constraint is: $8x \leq 50$ .

8x \leq 50 \\ x \leq 6.25

Since she can't buy a fraction of a ticket, the answer is 6 tickets.

SAT tip: Watch for clue words—"at least" means $\geq$ , "no more than" means $\leq$ .

Let's practice with a few College Board questions!

Question 1

Marisa needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for paying the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following systems of inequalities represents the conditions described if $x$ is the number of junior directors and $y$ is the number of senior directors?

Question 2

A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?

Question 3

Adam’s school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, $w$ , that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of $w$ for which it would be faster for Adam to walk to school?

✓

linear inequalities, mastered

You can now solve and interpret any linear inequality problem—one variable, two variables, word, and SAT-style trap!

Next Up

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simplify and manipulate algebraic expressions