Question
Justifying Algebraic Steps Worksheet
1. Fill in the blank with the appropriate property that justifies the step in solving the equation 2x + 8 = 16 :
2x + 8 = 16 2x = 8 (______ Property)
2. A carpenter is cutting boards for a project. The length of each board can be represented by the equation
3x − 5 = 22 , where x is the number of feet. Which property justifies the step from 3x − 5 = 22 to 3x = 27 ?
A. Distributive
Property
C. Multiplicative
Property of Equality
B. Additive Property
of Equality
D. Commutative
Property
3. Solve the inequality 4x + 2 ≤ 18 and justify each step using algebraic properties or properties of real numbers.
Explain how these properties ensure that the solution is valid for all real numbers that satisfy the inequality
1. Fill in the blank with the appropriate property that justifies the step in solving the equation 2x + 8 = 16 :
2x + 8 = 16 2x = 8 (______ Property)
2. A carpenter is cutting boards for a project. The length of each board can be represented by the equation
3x − 5 = 22 , where x is the number of feet. Which property justifies the step from 3x − 5 = 22 to 3x = 27 ?
A. Distributive
Property
C. Multiplicative
Property of Equality
B. Additive Property
of Equality
D. Commutative
Property
3. Solve the inequality 4x + 2 ≤ 18 and justify each step using algebraic properties or properties of real numbers.
Explain how these properties ensure that the solution is valid for all real numbers that satisfy the inequality
Answers
GPT-5 mini
1. 2x + 8 = 16 → 2x = 8 (Subtraction Property of Equality)
Explanation: 8 was subtracted from both sides (or use the Additive Property of Equality).
2. Answer: B. Additive Property of Equality
Explanation: 5 was added to both sides to remove −5 from the left, giving 3x = 27.
3. Solve and justify:
- 4x + 2 ≤ 18 (given)
- 4x ≤ 16 (subtract 2 from both sides — Addition/Subtraction Property of Inequality)
Justification: Real numbers are closed under subtraction and subtracting the same number from both sides preserves the inequality.
- x ≤ 4 (divide both sides by 4 — Multiplicative Property of Inequality / Division Property by a positive number)
Justification: 4 is positive, so dividing both sides by 4 does not change the direction of the inequality.
Solution: x ≤ 4, or in interval notation (−∞, 4].
Why these properties ensure validity: the additive and multiplicative properties (and the order axioms for real numbers) guarantee that performing the same additive or multiplicative operation on both sides gives an equivalent inequality. Closure, existence of additive inverses, and the rule that dividing/multiplying by a positive number preserves the order ensure every real number satisfying the original inequality also satisfies each transformed inequality and vice versa.
Explanation: 8 was subtracted from both sides (or use the Additive Property of Equality).
2. Answer: B. Additive Property of Equality
Explanation: 5 was added to both sides to remove −5 from the left, giving 3x = 27.
3. Solve and justify:
- 4x + 2 ≤ 18 (given)
- 4x ≤ 16 (subtract 2 from both sides — Addition/Subtraction Property of Inequality)
Justification: Real numbers are closed under subtraction and subtracting the same number from both sides preserves the inequality.
- x ≤ 4 (divide both sides by 4 — Multiplicative Property of Inequality / Division Property by a positive number)
Justification: 4 is positive, so dividing both sides by 4 does not change the direction of the inequality.
Solution: x ≤ 4, or in interval notation (−∞, 4].
Why these properties ensure validity: the additive and multiplicative properties (and the order axioms for real numbers) guarantee that performing the same additive or multiplicative operation on both sides gives an equivalent inequality. Closure, existence of additive inverses, and the rule that dividing/multiplying by a positive number preserves the order ensure every real number satisfying the original inequality also satisfies each transformed inequality and vice versa.