Practice questions for |ax + b| < c absolute value inequalities

📚 Topic Summary

Absolute value inequalities in the form $|ax + b| < c$ mean that the distance between $ax + b$ and 0 is less than $c$. This translates into two separate inequalities: $-c < ax + b < c$. To solve, you need to isolate $x$ in both inequalities, keeping in mind that dividing by a negative number flips the inequality sign. Remember to check your answers!

Solving $|ax + b| < c$ problems basically means "squishing" $ax+b$ between $-c$ and $c$.

🧮 Part A: Vocabulary

Match the term to its definition:

1. Term: Absolute Value
2. Term: Inequality
3. Term: Variable
4. Term: Constant
5. Term: Solution Set

1. Definition: A symbol representing an unknown value.
2. Definition: A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥.
3. Definition: A fixed value that does not change.
4. Definition: The distance of a number from zero.
5. Definition: The set of all values that satisfy an inequality.

📝 Part B: Fill in the Blanks

Complete the following sentences:

When solving the absolute value inequality $|ax + b| < c$, we rewrite it as __________ $ < ax + b < $ __________. To isolate $x$, we first __________ __________ from all parts of the inequality, and then __________ by __________. Remember to __________ the inequality sign if we divide by a __________ number.

🤔 Part C: Critical Thinking

Explain in your own words why we need to consider two separate inequalities when solving an absolute value inequality like $|ax + b| < c$. Use an example to illustrate your explanation.