Solve: −9 < 2x − 5 < 9 → −4 < 2x < 14 → −2 < x < 7.
Solving the Inequality −9 < 2x − 5 < 9: A Step-by-Step Guide
Learning how to solve inequalities step-by-step is essential for mastering algebra and building problem-solving skills. One common inequality students encounter is:
$$ -9 < 2x - 5 < 9 $$
This compound inequality can seem tricky at first, but breaking it down into smaller, manageable parts makes it easy to solve. In this article, we’ll walk through solving this inequality, explaining each step clearly and showing how to arrive at the final solution: $$ -2 < x < 7 $$
Understanding the Compound Inequality
The inequality $$ -9 < 2x - 5 < 9 $$ is a compound inequality involving a expression on both sides. It states that 2x minus 5 lies strictly between –9 and 9. To solve for x, we need to isolate x while maintaining the inequality’s truth.
Step 1: Add 5 to All Parts
To eliminate the constant term –5, add 5 to every part of the inequality:
$$ -9 + 5 < 2x - 5 + 5 < 9 + 5 $$
Simplifying each term:
$$ -4 < 2x < 14 $$
This step preserves the inequality because we’ve performed the same operation on all three parts.
Step 2: Divide All Parts by 2
Now isolate x by dividing every part of the inequality by 2:
$$ rac{-4}{2} < rac{2x}{2} < rac{14}{2} $$
Which simplifies to:
$$ -2 < x < 7 $$
Final Result
The solution to the inequality $$ -9 < 2x - 5 < 9 $$ is $$ -2 < x < 7 $$
Why This Matters
This type of inequality often appears in real-world problems involving ranges, restrictions, or constraints—such as temperature limits, budget thresholds, or performance metrics. Being able to solve such inequalities helps students interpret conditions and make precise conclusions.
Quick Recap: Solving −9 < 2x − 5 < 9
- Add 5 → $-4 < 2x < 14$
- Divide by 2 → $-2 < x < 7$
Mastering these steps gives you the confidence to tackle more complex inequalities with ease.
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Start solving inequalities with clarity and confidence—begin with breaking down compound expressions like this one, and watch your algebra skills grow!
