Studdy AI Logo
Math  /  Algebra

QuestionWhich statements are true? Check all that apply. The equation ∣−x−4∣=8|-x-4|=8 will have two solutions. The equation 3.4∣0.5x−42.1∣=−20.63.4|0.5 x-42.1|=-20.6 will have one solution. The equation ∣12x−34∣=0\left|\frac{1}{2} x-\frac{3}{4}\right|=0 will have no solutions. The equation ∣2x−10∣=−20|2 x-10|=-20 will have two solutions. The equation ∣0.5x−0.75∣+4.6=0.25|0.5 x-0.75|+4.6=0.25 will have no solutions. The equation ∣18x−1∣=5\left|\frac{1}{8} x-1\right|=5 will have infinitely many solutions.

Studdy Solution

STEP 1

1. Absolute value equations can have zero, one, two, or no solutions depending on the equation.
2. An absolute value equation of the form ∣A∣=B |A| = B has solutions if B≥0 B \geq 0 .
3. If B<0 B < 0 , the equation ∣A∣=B |A| = B has no solutions.
4. If B=0 B = 0 , the equation ∣A∣=B |A| = B has exactly one solution where A=0 A = 0 .

STEP 2

1. Analyze each equation individually to determine the number of solutions.
2. Apply the properties of absolute value equations to each statement.
3. Conclude which statements are true.

STEP 3

Analyze the equation ∣−x−4∣=8 |-x-4|=8 :
The equation ∣−x−4∣=8 |-x-4|=8 can be rewritten as two separate equations: −x−4=8or−x−4=−8 -x-4 = 8 \quad \text{or} \quad -x-4 = -8
Solving these:
1. −x−4=8-x-4 = 8 leads to −x=12 -x = 12 and thus x=−12 x = -12 .
2. −x−4=−8-x-4 = -8 leads to −x=−4 -x = -4 and thus x=4 x = 4 .
This equation has two solutions: x=−12 x = -12 and x=4 x = 4 .

STEP 4

Analyze the equation 3.4∣0.5x−42.1∣=−20.6 3.4|0.5 x-42.1|=-20.6 :
The absolute value equation 3.4∣0.5x−42.1∣=−20.6 3.4|0.5 x-42.1|=-20.6 has a negative value on the right side. Since absolute values cannot be negative, this equation has no solutions.

STEP 5

Analyze the equation ∣12x−34∣=0 \left|\frac{1}{2} x-\frac{3}{4}\right|=0 :
The equation ∣12x−34∣=0 \left|\frac{1}{2} x-\frac{3}{4}\right|=0 implies that the expression inside the absolute value must be zero: 12x−34=0 \frac{1}{2} x - \frac{3}{4} = 0
Solving for x x : 12x=34 \frac{1}{2} x = \frac{3}{4} x=34×2=32 x = \frac{3}{4} \times 2 = \frac{3}{2}
This equation has exactly one solution: x=32 x = \frac{3}{2} .

STEP 6

Analyze the equation ∣2x−10∣=−20 |2 x-10|=-20 :
The absolute value equation ∣2x−10∣=−20 |2 x-10|=-20 has a negative value on the right side. Since absolute values cannot be negative, this equation has no solutions.

STEP 7

Analyze the equation ∣0.5x−0.75∣+4.6=0.25 |0.5 x-0.75|+4.6=0.25 :
First, isolate the absolute value: ∣0.5x−0.75∣=0.25−4.6 |0.5 x-0.75| = 0.25 - 4.6 ∣0.5x−0.75∣=−4.35 |0.5 x-0.75| = -4.35
The absolute value equation has a negative value on the right side. Since absolute values cannot be negative, this equation has no solutions.

STEP 8

Analyze the equation ∣18x−1∣=5 \left|\frac{1}{8} x-1\right|=5 :
The equation ∣18x−1∣=5 \left|\frac{1}{8} x-1\right|=5 can be rewritten as two separate equations: 18x−1=5or18x−1=−5 \frac{1}{8} x - 1 = 5 \quad \text{or} \quad \frac{1}{8} x - 1 = -5
Solving these:
1. 18x−1=5\frac{1}{8} x - 1 = 5 leads to 18x=6 \frac{1}{8} x = 6 and thus x=48 x = 48 .
2. 18x−1=−5\frac{1}{8} x - 1 = -5 leads to 18x=−4 \frac{1}{8} x = -4 and thus x=−32 x = -32 .
This equation has two solutions: x=48 x = 48 and x=−32 x = -32 .
Conclusion: - The equation ∣−x−4∣=8 |-x-4|=8 will have two solutions. (True) - The equation 3.4∣0.5x−42.1∣=−20.6 3.4|0.5 x-42.1|=-20.6 will have one solution. (False) - The equation ∣12x−34∣=0 \left|\frac{1}{2} x-\frac{3}{4}\right|=0 will have no solutions. (False) - The equation ∣2x−10∣=−20 |2 x-10|=-20 will have two solutions. (False) - The equation ∣0.5x−0.75∣+4.6=0.25 |0.5 x-0.75|+4.6=0.25 will have no solutions. (True) - The equation ∣18x−1∣=5 \left|\frac{1}{8} x-1\right|=5 will have infinitely many solutions. (False)

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord