Solving Multi-Step Linear Inequalities
Solving multi-step linear inequalities follows the same sequence as solving multi-step equations — distribute, combine like terms, move variables to one side, move constants to the other, then divide — with one critical difference: when you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. For example, -2x + 3 < 9 becomes -2x < 6, then dividing by -2 flips the sign to give x > -3. Chapter 2 of OpenStax Elementary Algebra 2E covers this skill right after multi-step equations, reinforcing the parallel process while highlighting this essential sign-flip rule.
Key Concepts
Property To solve a multi step linear inequality, follow a systematic flow: 1. Simplify each side completely (distribute and combine like terms). 2. Use the Addition or Subtraction Properties of Inequality to collect all variable terms on one side and all constant terms on the other side. 3. Use the Multiplication or Division Properties of Inequality to isolate the variable. (Remember to reverse the inequality sign if you multiply or divide by a negative number!).
Examples Example 1: Solve $3x + 5 20$. Subtract 5 from both sides to get $3x 15$. Divide by 3 to get $x 5$. Example 2 (Variables on both sides): Solve $7p 2 \leq 3p + 10$. Subtract $3p$ from both sides to gather variables on the left: $4p 2 \leq 10$. Add 2 to both sides to gather constants on the right: $4p \leq 12$. Divide by 4 to get $p \leq 3$. Example 3 (Negative division): Solve $5(k 2) 20$. Distribute to get $5k 10 20$. Add 10 to both sides: $5k 10$. Divide by 5 to get $k 2$. (The sign stays the same because we divided by a positive 5).
Explanation Solving a multi step inequality uses the exact same strategy as solving a multi step equation: clean up both sides, move the letters to one team and the numbers to the other, and then isolate the variable. The only difference is the golden rule of inequalities—you must stay highly alert during the very last step. If you divide or multiply by a negative number to get the variable by itself, you must flip the inequality symbol.
Common Questions
How do you solve a multi-step linear inequality?
Use the same steps as a multi-step equation: distribute, combine like terms, isolate the variable using inverse operations. The only extra rule is: flip the inequality sign whenever you multiply or divide by a negative number.
When do you flip the inequality sign?
You flip the inequality sign only when multiplying or dividing both sides by a negative number. Adding, subtracting, or multiplying or dividing by a positive number does not change the direction.
What is the difference between solving an equation and solving an inequality?
Equations produce a single value or set of values. Inequalities produce a range of values. The process is nearly identical except for the sign-flip rule when dividing by a negative.
How do I write the solution to a linear inequality?
Write it as an inequality (x > 3), as a number line with an open or closed circle and an arrow, or in interval notation: (3, infinity) for x > 3.
When do students learn multi-step linear inequalities?
This is an algebra 1 skill covered in OpenStax Elementary Algebra 2E Chapter 2: Solving Linear Equations and Inequalities.
What is a common mistake when solving multi-step inequalities?
Forgetting to flip the inequality sign when dividing by a negative number. For example, dividing -2x < 6 by -2 gives x > -3, not x < -3.
How do I check my solution to a linear inequality?
Pick a value in your solution range and substitute it into the original inequality. If the result is a true statement, your solution is correct.