Study Guide and Intervention Solving Absolute Value Equations Absolute Value Expressions The Absolute Value of a Real Number Example Evaluate each expression. 1. 4 2. 40 3. -3 4. 0 Exercises Evaluate each expression. 1. -4 2. 6+1 3. x-1 4. 4a-12 5. -m-1 6. 7x-3 7. l-1 8. 10 / l-1 9. 14 / (x-4) 10. 16 / (2x-1) 11. 12-v-5 12. z-1 13. 12-5 14. 1-1 15. 4u-1 16. 40-4
The Correct Answer and Explanation is:
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Let’s go through the Pekioo 1-4 Study Guide and Intervention: Solving Absolute Value Equations — specifically the section on evaluating Absolute Value Expressions.
🔢 What is Absolute Value?
The absolute value of a real number is its distance from 0 on a number line, regardless of direction.
- It is always non-negative.
- Symbolically, it’s written as: ∣a∣={aif a≥0−aif a<0|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}∣a∣={a−aif a≥0if a<0
✅ Examples (From the Guide):
Evaluate each expression:
- ∣4∣=4|4| = 4∣4∣=4
- ∣40∣=40|40| = 40∣40∣=40
- ∣−3∣=3|-3| = 3∣−3∣=3
- ∣0∣=0|0| = 0∣0∣=0
✍️ Exercises with Answers:
- ∣−4∣=4|-4| = 4∣−4∣=4
- ∣6+1∣=∣7∣=7|6 + 1| = |7| = 7∣6+1∣=∣7∣=7
- ∣x−1∣|x – 1|∣x−1∣ → Cannot evaluate without value of x
- ∣4a−12∣|4a – 12|∣4a−12∣ → Cannot evaluate without value of a
- ∣−m−1∣|-m – 1|∣−m−1∣ = ∣−m−1∣|-m – 1|∣−m−1∣ → Depends on m
- ∣7x−3∣|7x – 3|∣7x−3∣ → Depends on x
- ∣−1∣=1|-1| = 1∣−1∣=1
- 10∣−1∣=101=10\frac{10}{|-1|} = \frac{10}{1} = 10∣−1∣10=110=10
- 14∣x−4∣\frac{14}{|x – 4|}∣x−4∣14 → Depends on x
- 16∣2x−1∣\frac{16}{|2x – 1|}∣2x−1∣16 → Depends on x
- ∣12−v−5∣=∣7−v∣|12 – v – 5| = |7 – v|∣12−v−5∣=∣7−v∣ → Depends on v
- ∣z−1∣|z – 1|∣z−1∣ → Depends on z
- ∣12−5∣=∣7∣=7|12 – 5| = |7| = 7∣12−5∣=∣7∣=7
- ∣1−1∣=∣0∣=0|1 – 1| = |0| = 0∣1−1∣=∣0∣=0
- ∣4u−1∣|4u – 1|∣4u−1∣ → Depends on u
- (|40 – 4| = |36| = 36**
🧠 Explanation
Absolute value expressions are a core concept in algebra. They represent the distance of a number from zero, without considering direction. For instance, both 3 and -3 have an absolute value of 3 because they are both three units from zero on the number line.
In solving these expressions, we follow standard order of operations (PEMDAS) first, then apply the absolute value. For example, in ∣6+1∣|6 + 1|∣6+1∣, we compute the inside first: 6+1=76 + 1 = 76+1=7, then take the absolute value: ∣7∣=7|7| = 7∣7∣=7.
When an expression contains variables like xxx, aaa, or zzz, we cannot evaluate without knowing the variable’s value. Instead, we simplify the expression as far as possible. For example, ∣x−1∣|x – 1|∣x−1∣ stays in that form unless xxx is known.
Expressions involving division like 10∣−1∣\frac{10}{|-1|}∣−1∣10 require first evaluating the absolute value in the denominator. This is important because absolute value ensures the denominator won’t be negative, helping avoid undefined results due to division by zero.
Understanding absolute value is also essential for solving equations such as ∣x∣=a|x| = a∣x∣=a. These equations split into two cases: x=ax = ax=a and x=−ax = -ax=−a, since both values have the same absolute value.
Overall, absolute value expressions teach us to handle both positive and negative numbers in a uniform way, and help lay the foundation for solving more complex algebraic equations later on.
