Algebra • 35% of test
Linear Inequalities
Select your difficulty level to start practicing. We recommend mastering each level before moving to the next.
Beginner5 Sets
Beginner Practice
Start here to build your foundation. Clear problems and straightforward calculations.
Target Score
400-550
Intermediate5 Sets
Intermediate Practice
Level up with more complex equations and multi-step problems.
Target Score
550-700
Advanced3 Sets
Advanced Practice
Master the hardest concepts. Complex word problems and abstract applications.
Target Score
700-800
What is Linear Inequalities?
Linear inequalities are solved like equations but with one crucial difference: when multiplying or dividing by a negative number, you must flip the inequality sign.
Step-by-Step Strategy
⚠️ Common Traps to Avoid
Frequently Asked Questions
When do I flip the inequality sign?
ONLY when multiplying or dividing by a negative number. Adding or subtracting never changes the direction.
What does 'at most' mean?
'At most' means ≤ (less than or equal to). 'At least' means ≥ (greater than or equal to).
How do I solve compound inequalities?
Treat it as two separate inequalities and find where the solutions overlap (AND) or combine (OR).
Can Desmos help with inequalities?
Yes, Desmos can shade regions for inequalities. Practice using this tool in our <a href='/math'>SAT Math practice section</a> to visualize range solutions.
How do I check my answer?
Pick a value in your solution range and substitute into the original inequality. It should be true.
Are inequality word problems common?
Yes, especially involving budgets, limits, or constraints. Look for words like 'maximum', 'minimum', 'at most'.
How do I graph an inequality on the coordinate plane?
Graph the boundary line first. Use a dashed line for < or > and a solid line for ≤ or ≥, then shade the appropriate region.
What happens if I multiply by 0 in an inequality?
You should never multiply an inequality by 0, as it destroys the relationship and makes it impossible to solve for the variable.
How do I solve 'double' or compound inequalities like 5 < 2x + 1 < 11?
Perform the same operation on all three parts simultaneously. Subtract 1, then divide all parts by 2.
What is the interval notation for x ≥ 5?
The interval would be [5, ∞). The bracket means 5 is included, and the parenthesis is always used for infinity.
Can an inequality have 'No Solution'?
Yes, for example, x + 5 < x + 2 simplifies to 5 < 2, which is false, so there is no solution.
Can an inequality have 'All Real Numbers' as a solution?
Yes, if it simplifies to a true statement like 5 > 2, then any value of x will satisfy the inequality.
What does the 'Absolute Value' inequality |x| < 5 mean?
It means the distance from zero is less than 5, which translates to -5 < x < 5.
How do I check a shaded region on Desmos?
Simply type the inequality (e.g., y < 2x + 3) and Desmos will automatically shade the correct side of the boundary.
What is the most common trap with negative variables?
Students often forget to flip the sign if they divide by -1. Master these sign rules with our <a href='/math/linear-inequalities/beginner'>Beginner Linear Inequalities sets</a>.