Linear inequalities in one or two variables fall under the Algebra category on the Digital SAT Math section (44 questions, 70 minutes). These questions ask you to build inequalities from word problems, interpret what parts of an inequality mean in context, check whether specific points satisfy an inequality or system, and move between algebraic, graphical, and tabular forms. The core algebra is similar to linear equations, but instead of a single answer you're working with a solution set. Key skills: translating constraint language, applying the sign-flip rule, and reading half-planes on the xy-plane.

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Building and Solving Linear Inequalities

Most inequality questions start with a real-world scenario where some quantity is limited. Your job is constraint modeling: turning English into algebra.

Translating Constraint Language

These phrases map directly to inequality symbols:

"At least" or "no fewer than" → $\geq$
"At most" or "no more than" → $\leq$
"More than" or "greater than" → $>$
"Less than" or "fewer than" → $<$

The distinction between "at least 200" ( $\geq 200$ , includes 200) and "more than 200" ( $> 200$ , excludes 200) is exactly the kind of detail the Digital SAT tests.

Solving One-Variable Inequalities

Solving works just like equations with one critical rule: flip the inequality sign when you multiply or divide both sides by a negative number.

Example 1 — Straightforward: A gym charges a $25 monthly fee plus $8 per class. A member wants to spend no more than $75 per month. Which inequality represents the number of classes $c$ the member can take?

Set up the total cost and apply the constraint:

$8c + 25 \leq 75$

Solve:

$8c \leq 50$

$c \leq 6.25$

Since $c$ must be a whole number, the member can take at most 6 classes.

Example 2 — Sign Flip: A company's profit is modeled by $P = -3x + 900$ , where $x$ is the number of units beyond a certain threshold. For what values of $x$ is the profit greater than $150?

$-3x + 900 > 150$

$-3x > -750$

Divide by $-3$ and flip the sign:

$x < 250$

The solution set is all values of $x$ less than 250.

Interpreting Constants, Variables, and Solutions in Context

Some questions don't ask you to solve anything. They give you a linear inequality and ask what a specific number or expression represents. This tests whether you understand the structure of the inequality, not just the mechanics.

Example 3: The inequality $12m + 35 \leq B$ models a delivery service, where $B$ is the customer's budget in dollars and $m$ is the number of miles. What does each part represent?

12 is the coefficient of $m$ , so it represents the charge per mile (dollars per mile).
35 is the constant term, independent of miles, so it represents the flat fee in dollars.
$B$ is the upper bound of the total cost, so it represents the maximum amount the customer is willing to spend.

Common trap: Answer choices will describe real quantities from the problem but attach them to the wrong part of the inequality. Always identify which term the question asks about and match its structural role (per-unit rate, fixed amount, total, limit).

Two-Variable Inequalities and the XY-Plane

When an inequality has two variables, its graph in the xy-plane is a half-plane: a boundary line with shading on one side. Every point in the shaded region (and on the line, if the inequality uses $\leq$ or $\geq$ ) is part of the solution set.

Graphing Inequalities

To graph $y \leq 2x + 3$ :

Graph the boundary line $y = 2x + 3$ . Use a solid line because the inequality includes "equal to" ( $\leq$ ). Use a dashed line for strict inequalities ( $<$ or $>$ ).
Shade below the line because $y$ is less than the expression. For $y \geq 2x + 3$ , shade above.

Quick check: plug in $(0, 0)$ . Is $0 \leq 2(0) + 3$ ? Yes, $0 \leq 3$ . So the side containing the origin is the solution region.

Interpreting Points

Example 4: A bakery models its labor constraint as $3c + b \leq 24$ , where $c$ is the number of cakes (3 hours each) and $b$ is the number of cookie batches (1 hour each), and 24 is the total hours available.

Does $(5, 10)$ satisfy the inequality?

$3(5) + 10 = 25 > 24 \quad \text{No}$

Making 5 cakes and 10 batches requires 25 hours, which exceeds the limit. On the graph, $(5, 10)$ falls outside the shaded region.

Does $(4, 12)$ satisfy it?

$3(4) + 12 = 24 \leq 24 \quad \text{Yes}$

This combination uses exactly the available hours and lands on the boundary line.

Systems of Inequalities

When a problem has multiple constraints, you get a system of inequalities. The solution set is the overlap of all the individual shaded regions in the xy-plane.

Example 5: A student works two part-time jobs. Job A pays $15/hr and Job B pays $12/hr. The student wants to earn at least $300 per week but can work no more than 25 total hours. Let $a$ = hours at Job A and $b$ = hours at Job B.

Earnings constraint:

$15a + 12b \geq 300$

Time constraint:

$a + b \leq 25$

(Implied: $a \geq 0$ and $b \geq 0$ , since hours can't be negative.)

Check whether $(10, 14)$ is in the solution set:

Earnings: $15(10) + 12(14) = 150 + 168 = 318 \geq 300$ ✓
Time: $10 + 14 = 24 \leq 25$ ✓

Both inequalities are satisfied, so $(10, 14)$ is a valid combination.

Connecting Representations: Tables, Graphs, and Algebra

The Digital SAT expects you to move between representations. You might see a table of values and need to identify which inequality fits, or see a graph and write the corresponding inequality.

From a table: Plug each $(x, y)$ pair into the answer choices. The correct inequality must hold for every point. Boundary points (where equality holds exactly) tell you whether to use $\leq$ / $\geq$ or $<$ / $>$ .

From a graph: Identify the boundary line's slope and y-intercept. Then check the direction of shading and whether the line is solid or dashed.

Solid line, shaded below → $y \leq mx + b$
Dashed line, shaded above → $y > mx + b$

Example 6: A graph shows a dashed line through $(0, 4)$ and $(2, 0)$ with shading below the line.

Find the slope:

$m = \frac{0 - 4}{2 - 0} = -2$

The y-intercept is 4, so the boundary line is $y = -2x + 4$ .

Dashed line (strict) + shading below:

$y < -2x + 4$

What to Watch For on Test Day

"At least" vs. "more than": These map to different symbols ( $\geq$ vs. $>$ ). Read constraint language carefully — the wrong symbol means the wrong answer.
Flipping the sign: When you multiply or divide by a negative number, the inequality reverses. This is the most common algebraic error on these questions.
Solid vs. dashed lines: A solid boundary line means points on the line are included ( $\leq$ or $\geq$ ). A dashed line means they're excluded ( $<$ or $>$ ). Check this before eliminating answer choices.
Testing points in systems: A point must satisfy every inequality in the system, not just one. Plug the point into each inequality separately.
Whole-number context answers: If the algebraic answer is a decimal, think about whether to round up or down based on the direction of the inequality and the real-world context. "At least 333.3 units" means you need 334 units, not 333.