Linear Inequalities on the SAT

Why the SAT Emphasizes Linear Inequalities

When a problem says a cost must stay under $40$ or a speed must be at least $50$ miles per hour, it is not asking for one perfect number. It is asking for a range of values that all work. That is exactly what inequalities describe, and the SAT loves them.

The hard part is not solving the inequality, it is representing the solution correctly. This lesson shows how to solve step by step, when to flip the inequality sign, and how to show the answer on a number line or with Desmos shading.

A Simple Definition Unlocks Linear Inequalities

An inequality describes a whole set of solutions. If you solve $x \geq -4$, every value greater than or equal to $-4$ works, not just one number. That is why a number line is often the clearest way to show the result.

The rule that causes the most errors is this, if you multiply or divide by a negative, the inequality sign flips. You can always check your work by testing a value that should satisfy the inequality and confirming it makes the original statement true.

How the SAT Tests Linear Inequalities

• Solve a one step or two step inequality and choose the correct interval.
• Translate a word problem into an inequality.
• Handle compound inequalities like $2 < x \leq 9$.
• Interpret a shaded region on a number line or coordinate plane.

Work Through Linear Inequalities Step by Step

Let us solve a typical SAT inequality and then interpret it as a range.

Use Desmos to Check Linear Inequalities

Desmos can shade the solution directly, which is helpful for compound inequalities or messy fractions. You can also graph both sides and see where one is below the other.

For simple one step inequalities, algebra is usually faster. Desmos is faster when you are dealing with multiple inequalities or decimals and want a quick visual check.

Expert move: Enter the inequality directly so Desmos shades the solution region. For systems, graph both and focus on the overlap; use a quick test point to confirm the shading.

When to skip Desmos: For one-step inequalities, solving by hand is faster; use Desmos only to verify.

• Desmos features used: inequality shading, graphing.
• Common mistake: forgetting that the solution is on the $x$ axis, not a single point.

Practice Linear Inequalities with SAT-Style Questions

Work these first, then compare your reasoning with the explanations.

Key Takeaways to Remember for Linear Inequalities

• Flip the inequality sign only when you multiply or divide by a negative.
• Use open circles for $<$ and $>$, closed circles for $\leq$ and $\geq$.
• Desmos is best for visual checks and compound inequalities.