Answer: The true statements are:
Statement 1: The equation |-x -4| = 8 will have two solutions
Statement 5: The equation | 0.5x - 0.75 | + 4.6 = 0.25 will have no solutions.
Explanation:
These are six completely different and independent statements. Every one has one equation that needs to be solved.
Which statements are true?
A. Statement 1: The equation |-x -4| = 8 will have two solutions
Answer: true
Justification:
1. First step is to remove the bars of the absolute value function:
⇒ - x - 4 = 8 or - x - 4 = - 8
2. Solve first equation: - x - 4 = 8
- addition property of equality (add 4): - x = 8 + 4
- combine like terms: - x = 12
- multiplication property of equality (multiply by -1): x = - 12
3. Solve second equation: - x - 4 = - 8
- addition property of equality (add 4): - x = -8 + 4
- combine like terms: - x = - 4
- multiplication propery (multiplicate by -1): x = 4
Hence, the equation has two solutions x = - 12 and x = 4
B. Statement 2: The equation 3.4|0.5x - 42.1| = - 20.6 will have one solution.
Answer: false.
Justification:
- The left hand side cannot be negative, because the absolute value is always positive, and the right hand side is negative, hence, the equality is false.
C. Statement 3: The equation | (1/2)x - 3/4 | = 0 has no solutions
Answer: false
Justification:
- Since 0 is not either positive or negative the only solution of that equation is (1/2x) - 3/4 = 0
- By the addition property: (1/2)x = 3/4
- By the multiplication property: x = 2(3)/4 = 3/2
Therefore, the equation has one solution.
D. Statement 4: |2x - 10 | = - 20 will have two solutions
Answer: false
Justification:
- The left hand side is cannot be negative, and the right hand side is negative, hence the equality can never be true.
E. Statement 5: The equation | 0.5x - 0.75 | + 4.6 = 0.25 will have no solutions.
Answer: true.
Justification:
- Since the absolute value cannot be negative, when you add it to 4.6, you will obtain a number greater than or equal to 4.6, hence the left hand side will never be equal to the right hand side (0.25), making the statement false.
F. Statement 6: the equation | (1/8)x - 1 | = 5 will have infinetely many solutions.
Answer: false:
Justification:
1. Remove the bars:
- | (1/8)x - 1 | = 5 ⇒ (1/8)x - 1 = 5 or (1/8)x - 1 = - 5
2. You do not need to solve the equations to see that you will obtain only two solutions. Hence, the statement is false.
