Example Card: Solving an Inequality with Variables on Both Sides
Solve inequalities with variables on both sides in Grade 9 algebra by collecting variables on one side, isolating them, and remembering to flip the inequality sign when multiplying or dividing by a negative.
Key Concepts
Getting variables on one side is the first step to taming these tricky inequalities.
Example Problem.
Solve the inequality $3x + 5 4x + 26$.
Common Questions
How do you solve an inequality with variables on both sides?
Collect all variable terms on one side by adding or subtracting. Then isolate the variable using inverse operations. For 3x + 5 > -4x + 26: add 4x to both sides (7x + 5 > 26), subtract 5 (7x > 21), divide (x > 3).
When and why do you flip the inequality sign?
Flip the inequality sign ONLY when you multiply or divide both sides by a negative number. For example, -2x < 8 becomes x > -4 after dividing by -2 and flipping the sign. Adding or subtracting (even negatives) never requires a flip.
How do you check the solution to an inequality with variables on both sides?
Pick a test value that satisfies your solution. For x > 3, test x = 4: check 3(4) + 5 > -4(4) + 26: 17 > 10 ✓. Also test a value that should NOT work, like x = 2: check 3(2)+5 > -4(2)+26: 11 > 18 ✗. Both checks confirm x > 3.