Solving Multi-Step Compound Inequalities
Solve multi-step compound inequalities in Grade 9 algebra by applying inverse operations to all parts simultaneously for AND inequalities, or solving each part separately for OR inequalities.
Key Concepts
New Concept A compound inequality is made of two inequalities joined by the word AND or OR. For example, $ 2 < x \operatorname{AND} x \le 5$. What’s next Next, you’ll apply familiar algebraic properties to isolate the variable and find the solution set for these more complex statements.
Common Questions
How do you solve a multi-step AND (conjunction) compound inequality?
Apply the same inverse operations to all three parts simultaneously, maintaining the inequality signs. For -3 < 2x + 1 ≤ 9: subtract 1 from all parts (-4 < 2x ≤ 8), then divide by 2 (-2 < x ≤ 4).
How do you solve a multi-step OR (disjunction) compound inequality?
Solve each inequality separately using standard techniques, then combine the solution sets with union. For 3x + 1 < -5 OR 2x - 3 > 7: solve to get x < -2 OR x > 5. The solution is all values less than -2 or greater than 5.
When do you flip the inequality signs in a multi-step compound inequality?
Flip ALL inequality signs in the compound expression only when multiplying or dividing all parts by a negative number. For -3 < -2x ≤ 6, dividing by -2 flips both signs: 3/2 > x ≥ -3, rewritten as -3 ≤ x < 3/2.