Linear Inequalities in One or Two Variables

Linear inequalities show up on the Digital SAT in roughly 3–5 questions per test, spanning both modules. These questions ask you to build inequalities from word problems, interpret what parts of an inequality mean in context, determine whether specific points satisfy an inequality or system, and move between algebraic, graphical, and tabular forms. The core algebra is similar to working with linear equations, but instead of finding a single answer, you're working with a solution set of values that make the inequality true. Mastering this topic means knowing how to translate constraints into math, solve correctly (including the sign-flip rule), and read graphs in the xy-plane.

Building and Solving Linear Inequalities

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

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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 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.

Worked 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.

Worked 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. Instead, 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.

Worked Example 3: A delivery service charges a flat fee plus a per-mile rate. The inequality $12m + 35 \leq B$ models the situation where $B$ is the customer's budget in dollars and $m$ is the number of miles. What does the number 12 represent?

Look at where 12 sits in the expression: it's the coefficient of $m$, the per-mile variable. So 12 represents the charge, in dollars, per mile of delivery.

What about 35? It's the constant term, independent of miles. It represents the flat fee in dollars.

And $B$? The inequality says the total cost must be less than or equal to $B$, so $B$ is 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 check which term or factor 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 includes $\leq$ or $\geq$) is part of the solution set.

Graphing Inequalities

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

1. 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 $>$).
2. Shade below the line because $y$ is less than the expression. If the inequality were $y \geq 2x + 3$, you'd shade above.

A quick test: pick a point not on the line (the origin $(0, 0)$ is usually easiest). Plug it in: $0 \leq 2(0) + 3$ gives $0 \leq 3$, which is true. So the side containing $(0, 0)$ is the solution region.

Interpreting Points

Worked Example 4: A bakery makes cakes and cookies. Each cake requires 3 hours and each batch of cookies requires 1 hour. The bakery has at most 24 hours of labor available per day. The inequality $3c + b \leq 24$ models this constraint, where $c$ is the number of cakes and $b$ is the number of cookie batches.

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

$3(5) + 10 = 15 + 10 = 25$

Since $25 > 24$, the point $(5, 10)$ does not satisfy the inequality. In context, making 5 cakes and 10 batches of cookies would require 25 hours, which exceeds the 24-hour limit.

Does the point $(4, 12)$ satisfy it?

$3(4) + 12 = 12 + 12 = 24$

Since $24 \leq 24$, yes. This combination uses exactly the available hours.

On the graph, $(5, 10)$ would fall outside the shaded region, while $(4, 12)$ would land on the boundary line.

Systems of Inequalities

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

Worked Example 5: A student works two part-time jobs. Job A pays $\$15$ per hour and Job B pays $\$12$ per hour. The student wants to earn at least $\$300$ per week but can work no more than 25 total hours. If $a$ is hours at Job A and $b$ is hours at Job B, which system models these constraints?

Earnings constraint (at least $\$300$):

$15a + 12b \geq 300$

Time constraint (no more than 25 hours):

$a + b \leq 25$

You'd also have the implied constraints $a \geq 0$ and $b \geq 0$ since hours can't be negative.

On the xy-plane, the solution set is the region where both shaded areas overlap. Any point in that region represents a valid combination of hours. A point like $(10, 14)$ would need to be checked against both inequalities:

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

So $(10, 14)$ is in the solution set.

Connecting Representations: Tables, Graphs, and Algebra

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

From a table: If you're given several $(x, y)$ pairs and asked which inequality they satisfy, plug each pair into the answer choices. The correct inequality must be true for every point in the table. Pay attention to boundary points where the inequality is exactly equal, as this tells you whether to use $\leq$ / $\geq$ versus $<$ / $>$.

From a graph: Identify the boundary line's slope and y-intercept to write the equation of the line. Then determine the direction of shading (above or below) and whether the line is solid or dashed. A solid line shaded below means $y \leq mx + b$. A dashed line shaded above means $y > mx + b$.

Worked Example 6: A graph in the xy-plane shows a dashed line passing through $(0, 4)$ and $(2, 0)$ with shading below the line. What inequality does this represent?

Find the slope: $m = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2$

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

The line is dashed (strict inequality) and shading is below ($y$ is less than), so:

$y < -2x + 4$

What to Watch For on Test Day

1. "At least" vs. "more than": These map to different symbols ($\geq$ vs. $>$). Read constraint language carefully. Wrong symbol = wrong answer.

2. Flipping the sign: When you multiply or divide by a negative number, the inequality reverses. This is the single most common algebraic error on inequality questions.

3. Solid vs. dashed lines: On graphing inequalities questions, a solid boundary line means the points on the line are included ($\leq$ or $\geq$). A dashed line means they're not ($<$ or $>$). Check this before eliminating answer choices.

4. Testing points in systems: When a question asks whether a point satisfies a system of inequalities, it must satisfy every inequality in the system, not just one. Plug the point into each inequality separately.

5. Whole-number answers: In context problems (number of items, people, hours), your algebraic answer might be a decimal. Think about whether to round up or down based on the direction of the inequality. "At least 333.3 units" means you need 334 units, not 333.