Respuesta :
Answer:
1. The equation |-x -4| = 8 will have two solutions. (True)
5. The equation |0.5x – 0.75| + 4.6 = 0.25 will have no solutions. (True)
Step-by-step explanation:
1. The equation |-x -4| = 8 will have two solutions. (True)
|-x - 4| = 8
-x -4 = ±8
-x -4 = 8 and -x -4 = -8
-x = 8 + 4 and -x = -8 + 4
-x = 12 and -x = -4
x = -12 and x = 4
Therefore, it has two solutions x ∈ {-12, 4}
2. The equation 3.4|0.5x - 42.1| = -20.6 will have one solution. (False)
Since the right side of the equation has a negative value therefore, according to rules of absolute value equations, it has no solution.
3. The equation |1/2x - 3/4| = 0 will have no solutions. (False)
|(1/2)x - 3/4| = 0
(1/2)x - 3/4 = ±0
Since ±0 is the same
(1/2)x = 3/4
x = 2*3/4
x = 3/2
Therefore, it has one solution x = 3/2
4. The equation |2x – 10| = –20 will have two solutions. (False)
Since the right side of the equation has a negative value therefore, according to rules of absolute value equations, it has no solution.
5. The equation |0.5x – 0.75| + 4.6 = 0.25 will have no solutions. (True)
|0.5x – 0.75| + 4.6 = 0.25
|0.5x – 0.75| = 0.25 - 4.6
|0.5x – 0.75| = -4.35
Since the right side of the equation has a negative value therefore, according to rules of absolute value equations, it has no solution.
6. The equation |(1/8)x - 1| = 5 will have infinitely many solutions. (False)
|(1/8)x - 1| = 5
(1/8)x - 1 = ±5
(1/8)x - 1 = 5 and (1/8)x - 1 = -5
(1/8)x = 5 + 1 and (1/8)x = -5 + 1
(1/8)x = 6 and (1/8)x = -4
x = 6*8 and x = -4*8
x = 48 and x = -32
Therefore, it has two solutions x ∈ {48, -32}
The True statements are:
1. The equation |-x -4| = 8 will have two solutions.
5. The equation |0.5x – 0.75| + 4.6 = 0.25 will have no solutions.
We have Statements:
1. The equation |-x -4| = 8 will have two solutions. (True)
|-x - 4| = 8
-x -4 = ±8
-x -4 = 8 and -x -4 = -8
-x = 8 + 4 and -x = -8 + 4
-x = 12 and -x = -4
x = -12 and x = 4
So, it has two solutions.
The second statement is:
2. The equation 3.4|0.5x - 42.1| = -20.6 will have one solution.
As we can see that the right side of the equation has a negative value.
so, this equation has no solution.
The third statement is:
3. The equation[tex]|\frac{1}{2} x - \frac{3}{4} | = 0[/tex] will have no solutions.
[tex]|\frac{1}{2} x - \frac{3}{4} | = 0\\\frac{1}{2} x - \frac{3}{4}= ±0\\\frac{1}{2} x = \frac{3}{4}\\x = \frac{3}{2}[/tex]
So, it has one solution.
The fourth statement is:
4. The equation |2x – 10| = –20 will have two solutions.
As the right side of the equation has a negative value.
So, it has no solution.
The fifth statement is:
5. The equation |0.5x – 0.75| + 4.6 = 0.25 will have no solutions.
|0.5x – 0.75| + 4.6 = 0.25
|0.5x – 0.75| = 0.25 - 4.6
|0.5x – 0.75| = -4.35
As the right side of the equation has a negative value.
So, it has no solution.
The sixth statement is:
6. The equation[tex]|\frac{1}{8} x - 1| = 5[/tex] will have infinitely many solutions.
[tex]|\frac{1}{8} x - 1| = 5\\\frac{1}{8} x - 1 = \pm5[/tex]
[tex]\frac{1}{8} x - 1 = 5 , \frac{1}{8} x - 1 = -5\\\frac{1}{8} x = 5 + 1 , \frac{1}{8} x = -5 + 1\\\frac{1}{8}x = 6 , \frac{1}{8}x = -4\\x = 48 , x = -32[/tex]
So, it has two solutions.
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