Solve like an equation
Use inverse operations and preserve balance. Most equation-solving habits transfer directly.
Example2x + 1 < 9 → 2x < 8 → x < 4.
Chapter 03 of 10
Inequalities
What changes when an algebra problem has many possible answers?
An inequality compares quantities instead of declaring them equal. Its solution is usually an interval containing many numbers, so the answer is represented with inequality notation, interval notation, or a number-line graph.
The big ideas
Understand these before you worry about speed.
Reverse for a negative
Multiplying or dividing both sides by a negative reverses the inequality sign because it reverses the order of the numbers.
Example−3x ≥ 12 becomes x ≤ −4.
Keep compound inequalities balanced
A compound inequality such as a < x < b makes two comparisons at once. Perform the same operation on all three parts so the middle expression becomes isolated. If you multiply or divide by a negative, reverse both inequality signs.
Example becomes after subtracting from all three parts.
Graph the boundary
Use an open circle for < or > and a closed circle for ≤ or ≥. Shade toward values that satisfy the statement.
Examplex ≥ 2 uses a closed circle at 2 and shading right.
Worked example
Follow the reason for each step—not just the symbols.
Problem
Solve −2(3x − 4) > 14.
- Distribute: −6x + 8 > 14.
- Subtract 8: −6x > 6.
- Divide by −6 and reverse the sign: x < −1.
Answerx < −1
Try it yourself
Work on paper first. Open each answer only when you are ready to check.
1
Solve 5x − 3 ≤ 17.
Check answer
x ≤ 42
Solve −4x + 1 < 13.
Check answer
x > −33
Solve 2 < x + 5 ≤ 9.
Check answer
−3 < x ≤ 44
Write the interval for x ≥ −2.
Check answer
[−2, ∞)Practice and tools
Use the resource that matches what you need next.
Worksheet
Solving Inequalities
Practice sign reversal, compound inequalities, and absolute value.
Open resource WorksheetMiddle School Inequalities
Use a gentler set when the fundamentals need reinforcement.
Open resource QuizAlgebra 1 Readiness
Check prerequisite fluency with signed numbers and equations.
Open resource