MTH101 Lesson 2 Absolute Value from – Calculus and Analytical Geometry

KNOWLEDGE LEVEL (25 Questions)

Q1. What does the term absolute value mean?
Answer: Absolute value means the distance of a number from zero on the number line, always positive or zero.

Q2. What symbol is used for absolute value?
Answer: The symbol is two vertical bars, like this: |x|.

Q3. Find the absolute value of –9.
Answer: |–9| = 9
Because absolute value removes the negative sign.

Q4. What is |0| equal to?
Answer: |0| = 0
Zero is neither positive nor negative.

Q5. Is the absolute value of a number ever negative?
Answer: No, it’s always zero or positive, because distance cannot be negative.

Q6. What is |7| + |–2|?
Answer: 7 + 2 = 9
Take the positive values and add them.

Q7. What is |–3| – |–5|?
Answer: 3 – 5 = –2
Take absolute values first, then subtract.

Q8. If |x| = 8, what are the possible values of x?
Answer: x = 8 or x = –8
Both numbers are 8 units from zero.

Q9. What is the smallest possible value of |x|?
Answer: 0, which happens only when x = 0.

Q10. Why is |–a| = |a|?
Answer: Because absolute value ignores sign; both represent the same distance from zero.

Q11. Explain what |x – y| represents.
Answer: It shows the distance between x and y on the number line.

Q12. What is the geometric shape of the graph of y = |x|?
Answer: It forms a V-shape with the lowest point (vertex) at the origin (0,0).

Q13. What happens to the graph of y = |x| when we write y = |x – 3|?
Answer: The graph shifts 3 units to the right.

Q14. What happens to y = |x| when written as y = |x| + 2?
Answer: The graph moves 2 units up.

Q15. What does the equation |x| = 4 mean on a number line?
Answer: It means x is 4 units away from zero, so x = 4 or x = –4.

Q16. Why can |x| be written as a piecewise function?
Answer: Because |x| = x when x ≥ 0, and |x| = –x when x < 0.

Q17. If |a| < |b|, what can we say about a and b?
Answer: The number a is closer to zero than b on the number line.

Q18. Solve |x| = 6.
Answer: x = 6 or x = –6.
Both are 6 units from zero.

Q19. Solve |x – 4| = 2.
Answer: x – 4 = 2 or x – 4 = –2 → x = 6 or x = 2.

Q20. Solve |x + 3| = 5.
Answer: x + 3 = 5 or x + 3 = –5 → x = 2 or x = –8.

Q21. Solve |2x – 1| = 3.
Answer: 2x – 1 = 3 or 2x – 1 = –3 → x = 2 or x = –1.

Q22. Write the solution of |x| < 4 in interval form.
Answer: –4 < x < 4 → Interval: (–4, 4).

Q23. Write the solution of |x – 5| ≤ 3 in interval form.
Answer: –3 ≤ x – 5 ≤ 3 → 2 ≤ x ≤ 8 → Interval: [2, 8].

Q24. Solve |x + 2| > 7.
Answer: x + 2 > 7 or x + 2 < –7 → x > 5 or x < –9.

Q25. Find the distance between points –3 and 5 using absolute value.
Answer: |–3 – 5| = |–8| = 8
The distance is 8 units.