Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
Solving and Graphing Compound Inequalities
By Grade 7 at Think Academy (Grade 8 in school), students work with compound inequalities that combine two conditions using "and" or "or." The challenge is not the arithmetic but remembering the rules—many forget to flip the inequality sign when dealing with negatives, confuse "and" versus "or," struggle to combine results correctly, or misrepresent the solution on number lines. This guide walks step by step through solving and graphing so those common mistakes don't block progress.
What Is a Compound Inequality?
A compound inequality is when two inequalities are joined together by the words "and" or "or".
AND Compound Inequality
It means both conditions must be true at the same time.
Example:
This says that 𝑥 is greater than 1 AND less than 3.
OR Compound Inequality
It means at least one of the conditions must be true.
Example:
This says that 𝑥 is less than 4 OR greater than 8.
How to Solve and Graph Compound Inequalities
Let's solve the following example step by step:
−12 < 5𝑥 + 3 ≤ 23
Step 1: Break into Two Inequalities
−12 < 5𝑥 + 3 and 5𝑥 + 3 ≤ 23
Step 2: Solve Each Inequality Separately
For the first inequality,
−12 < 5𝑥 + 3
Subtract 3 from both sides:
−12 − 3 < 5𝑥 + 3 −3
−15 < 5𝑥
Divide both sides by the coefficient 5:
[\frac{-15}{5} < \frac{5x}{5}]
−3 < 𝑥
Keep variables left, constants right:
𝑥 > −3
Then, for the second inequality,
5𝑥 + 3 ≤ 23
Subtract 3 from both sides:
5𝑥 + 3 − 3 ≤ 23 − 3
5𝑥 ≤ 20
Divide both sides by the coefficient 5:
[\frac{5x}{5} \leq \frac{20}{5}]
𝑥 ≤ 4
Step 3: Combine the Results
Since -12 < 5𝑥 + 3 ≤ 23 is an "and" compound inequality, the solution is:
𝑥 > −3 and 𝑥 ≤ 4
−3 < 𝑥 ≤ 4
Step 4: Graph the Solution on the Number Line
- Put an open circle at -3, because it's greater than, not equal.
- Put a closed circle at 4, because it's less than or equal to.
- Shade the line between -3 and 4.
Example Problems: Solving and Graphing Compound Inequalities
- Solve the compound inequality and graph the solutions on the number line.
−13 < −3𝑥 + 2 ≤ 20
Answer:
Step 1: Split the compound inequality
−13 < −3𝑥 + 2 and −3𝑥 + 2 ≤ 20
Step 2: Solve each inequality separately
For −13 < −3𝑥 + 2:
Subtract 2 from both sides: −15 < −3𝑥
Divide both sides by −3 and flip sign:
[\frac{-15}{-3} > \frac{-3x}{-3}]
𝑥 < 5
For −3𝑥 + 2 ≤ 20:
Subtract 2: −3𝑥 ≤ 18
Divide by −3 and flip sign: 𝑥 ≥ −6
Step 3: Combine the results: −6 ≤ 𝑥 < 5
Step 4: Graph the solution on the number line
- Solve the compound inequality and graph the solutions on the number line.
−3𝑥 + 2 < 5 or −2𝑥 −1 ≥ 7
Answer:
For −3𝑥 + 2 < 5: subtract 2, divide by −3 and flip → 𝑥 > −1
For −2𝑥 −1 ≥ 7: add 1, divide by −2 and flip → 𝑥 ≤ −4
Step 3: Combine: 𝑥 > −1 or 𝑥 ≤ −4
Step 4: Graph the solution on the number line
Summary: Solving and Graphing Compound Inequalities
Compound inequalities let us describe ranges of possible values. The key steps are:
- Break the compound inequality into two simple inequalities.
- Solve each inequality.
- Combine the results using "and" or "or".
- Graph the solution on a number line.
Mastering these steps makes it much easier to see and understand the solution set visually. Always check your final answer to make sure it makes sense!
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