Study Guide

Linear Equations and Inequalities

These problems involve equations and inequalities with one or more variables, often requiring you to solve, graph, or interpret them in context.

Key Rules

Inequality rules mirror equation rules EXCEPT: multiplying or dividing by a negative flips the inequality sign.
Linear inequalities have a range of solutions, not a single value.
Systems of equations: use substitution or elimination to find the intersection point.
For word problems with inequalities, translate phrases like "at most," "at least," "no more than" carefully.

Common Mistakes to Avoid

Forgetting to flip the inequality when multiplying/dividing by a negative number.
"At least 5" means ≥ 5, not > 5.
In systems of equations, substituting before fully isolating a variable.
Choosing the wrong sign in a systems-of-equations word problem.

SAT Strategy Tips

→"At most" = ≤ | "at least" = ≥ | "more than" = > | "fewer than" = <
→For systems: if asked for x + y (not x and y separately), try adding or subtracting the equations directly.
→Graph inequalities mentally: shade above the line for > and ≥, below for < and ≤.
→The SAT loves asking for the minimum or maximum value of an expression — set up the inequality and solve.

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