Solving Linear Inequalities: Step-by-Step Guide
An inequality compares two things with a symbol like <, >, <=, or >= instead of an equals sign. You solve it almost the same way as an equation. There is one big difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign. The answer is usually a whole range of numbers, which we can write with interval notation or show on a number line.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Simplify both sides. Multiply out any parentheses and combine like terms on each side.\[k(ax+b) = kax + kb\]
- Move the x terms and numbers apart. Add or subtract to get the x terms on one side and the plain numbers on the other. Adding and subtracting never flip the sign.\[ax + b < cx + d \;\Rightarrow\; (a-c)x < d - b\]
- Get x alone and watch the sign. Divide both sides by the number in front of x. If that number is negative, flip the inequality sign.\[x < \frac{d-b}{a-c}\;\;\text{(flip if dividing by a negative)}\]
- Write the solution. Write it as an inequality, in interval notation, or on a number line. Use parentheses for < and >, and brackets for <= and >=.\[x < a \;\Leftrightarrow\; (-\infty, a)\]
- Check with test numbers. Pick one number that should be a solution and one that should not. Put each into the original inequality to see if it works.\[\text{Test } x_{\text{in}}\text{ and }x_{\text{out}}\]
Worked example
Solve and write in interval notation: -3x + 5 < 14
Step 1: There are no parentheses, so the inequality stays -3x + 5 < 14.
Step 2: Subtract 5 from both sides: -3x < 9.
Step 3: Divide both sides by -3. Because we divided by a negative number, flip the sign: x > -3.
Step 4: In interval notation, this is (-3, infinity).
Step 5: The answer is x > -3.
Check: Try x = 0, which should work: -3(0) + 5 = 5, and 5 < 14 is true. Try x = -5, which should not work: -3(-5) + 5 = 20, and 20 < 14 is false. Both tests agree.
Tips and common mistakes
- Flip the sign when you multiply or divide by a negative number. This is the most common mistake.
- Adding or subtracting never flips the sign, even when you add or subtract a negative number.
- Use parentheses for < and >, because the number is not included. Use brackets for <= and >=, because it is included.
- Infinity always gets a parenthesis, never a bracket.
- Test numbers are a quick way to check an inequality.
- On a number line, use an open circle for < and >, and a closed (filled) circle for <= and >=.
- If the x terms disappear, look at what is left. If it is true (like 3 < 5), every number works. If it is false, no number works.