Linear inequalities — Digital SAT Math practice
Solve, graph and interpret inequalities.
Key terms
- Strict Inequality: Uses < or >; graphed with open circles on number lines or dashed boundary lines.
- Inclusive Inequality: Uses ≤ or ≥; graphed with closed circles on number lines or solid boundary lines.
- Boundary Line: The line y = mx + b that separates the solution region from the non-solution region.
- Shaded Region: The area containing all coordinate pairs that satisfy the inequality when tested.
- System Solution: The overlapping shaded area where every point satisfies all inequalities simultaneously.
Common traps
- Forgetting to reverse the inequality sign when multiplying or dividing by a negative number changes the entire solution set.
- Drawing a solid line for strict inequalities like < or > instead of a dashed line.
- Using the origin as the test point when the boundary line passes through it; pick another point.
- Reading "at least" as strictly greater than instead of greater than or equal to.
Key ideas to know
Flipping the Inequality Sign
Solve for x: −4x + 8 > 20. A student writes: −4x > 12, so x > −3. What is wrong with this step?
The student forgot to flip the inequality sign when dividing by −4. Dividing by a negative reverses the direction, so x < −3.
Watch out: Always reverse < to > or > to < whenever you multiply or divide both sides by a negative number.
Solid vs dashed boundary
To graph y ≤ 3x − 1, should the boundary line be solid or dashed, and are points on the line part of the solution?
Solid. The symbol ≤ includes equality, so points exactly on the line satisfy the inequality.
Watch out: Mixing up strict symbols (< and >) with inclusive ones (≤ and ≥), which changes whether the boundary belongs to the solution.
Translating Word Problems to Inequalities
Translate the sentence into an inequality: "The number of tickets sold, t, must be at least 50 but no more than 200."
50 ≤ t ≤ 200. 'At least' means greater than or equal to, and 'no more than' (or 'at most') means less than or equal to.
Watch out: Mixing up 'more than' (>) with 'at least' (≥) or treating 'at most' as strictly less than (<).
Finding Solutions to Inequality Systems
You are solving the system y > x + 2 and y ≤ −x + 6. How do you identify which region contains all valid solutions?
Graph both boundary lines, shade above the first line and below or on the second, then find the overlapping shaded area. Every point in that overlap works.
Watch out: Assuming any single shaded region counts; only the intersection where all inequalities are true simultaneously is the solution.
Sample questions
Question 1
Which of the following describes all values of x for which 5 - 2x > 1?
- x > 2
- x < 3
- x < 2
- x > -2
Show answer and explanation
Answer: x < 2
5 - 2x > 1 gives -2x > -4; dividing by -2 flips the sign: x < 2.