Solved on Jan 17, 2024

Solve the absolute value equations: $|v-1|=7$ , $\left|\frac{y}{4}\right|=2$ , $|-10 k|=30$ , $|n+8|=1$ .

STEP 1

Assumptions
1. We are given four absolute value equations.
2. For each equation, we need to find the solutions for the variables $v$ , $y$ , $k$ , and $n$ .
3. The absolute value of a number is the distance of that number from zero on the real number line, which is always non-negative.
4. The equation $|x| = a$ has two solutions: $x = a$ and $x = -a$ if $a \geq 0$ .

STEP 2

Solve the first equation $|v-1|=7$ .
An absolute value equation $|x| = a$ has two solutions: $x = a$ and $x = -a$ . Therefore, we have two cases for $v-1$ :
1. $v-1 = 7$
2. $v-1 = -7$

STEP 3

Solve for $v$ in the first case of the first equation.
$v - 1 = 7$

STEP 4

Add 1 to both sides of the equation.
$v = 7 + 1$

STEP 5

Calculate the value of $v$ for the first case.
$v = 8$

STEP 6

Solve for $v$ in the second case of the first equation.
$v - 1 = -7$

STEP 7

Add 1 to both sides of the equation.
$v = -7 + 1$

STEP 8

Calculate the value of $v$ for the second case.
$v = -6$

STEP 9

The solutions for the first equation are $v = 8$ and $v = -6$ .

STEP 10

Solve the second equation $\left|\frac{y}{4}\right|=2$ .
Again, we have two cases for $\frac{y}{4}$ :
1. $\frac{y}{4} = 2$
2. $\frac{y}{4} = -2$

STEP 11

Solve for $y$ in the first case of the second equation.
$\frac{y}{4} = 2$

STEP 12

Multiply both sides of the equation by 4.
$y = 2 \times 4$

STEP 13

Calculate the value of $y$ for the first case.
$y = 8$

STEP 14

Solve for $y$ in the second case of the second equation.
$\frac{y}{4} = -2$

STEP 15

Multiply both sides of the equation by 4.
$y = -2 \times 4$

STEP 16

Calculate the value of $y$ for the second case.
$y = -8$

STEP 17

The solutions for the second equation are $y = 8$ and $y = -8$ .

STEP 18

Solve the third equation $|-10 k|=30$ .
Again, we have two cases for $-10k$ :
1. $-10k = 30$
2. $-10k = -30$

STEP 19

Solve for $k$ in the first case of the third equation.
$-10k = 30$

STEP 20

Divide both sides of the equation by -10.
$k = \frac{30}{-10}$

STEP 21

Calculate the value of $k$ for the first case.
$k = -3$

STEP 22

Solve for $k$ in the second case of the third equation.
$-10k = -30$

STEP 23

Divide both sides of the equation by -10.
$k = \frac{-30}{-10}$

STEP 24

Calculate the value of $k$ for the second case.
$k = 3$

STEP 25

The solutions for the third equation are $k = -3$ and $k = 3$ .

STEP 26

Solve the fourth equation $|n+8|=1$ .
Again, we have two cases for $n+8$ :
1. $n+8 = 1$
2. $n+8 = -1$

STEP 27

Solve for $n$ in the first case of the fourth equation.
$n + 8 = 1$

STEP 28

Subtract 8 from both sides of the equation.
$n = 1 - 8$

STEP 29

Calculate the value of $n$ for the first case.
$n = -7$

STEP 30

Solve for $n$ in the second case of the fourth equation.
$n + 8 = -1$

STEP 31

Subtract 8 from both sides of the equation.
$n = -1 - 8$

STEP 32

Calculate the value of $n$ for the second case.
$n = -9$
The solutions for the equations are: 1) $v = 8$ and $v = -6$ 2) $y = 8$ and $y = -8$ 3) $k = -3$ and $k = 3$ 4) $n = -7$ and $n = -9$

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Solve the absolute value equations: ∣v−1∣=7|v-1|=7∣v−1∣=7, ∣y4∣=2\left|\frac{y}{4}\right|=2∣∣​4y​∣∣​=2, ∣−10k∣=30|-10 k|=30∣−10k∣=30, ∣n+8∣=1|n+8|=1∣n+8∣=1.

STEP 1

STEP 2

Solve the first equation ∣v−1∣=7|v-1|=7∣v−1∣=7.An absolute value equation ∣x∣=a|x| = a∣x∣=a has two solutions: x=ax = ax=a and x=−ax = -ax=−a. Therefore, we have two cases for v−1v-1v−1:1. v−1=7v-1 = 7v−1=72. v−1=−7v-1 = -7v−1=−7

STEP 3

Solve for vvv in the first case of the first equation.v−1=7v - 1 = 7v−1=7

STEP 4

Add 1 to both sides of the equation.v=7+1v = 7 + 1v=7+1

STEP 5

Calculate the value of vvv for the first case.v=8v = 8v=8

STEP 6

Solve for vvv in the second case of the first equation.v−1=−7v - 1 = -7v−1=−7

STEP 7

Add 1 to both sides of the equation.v=−7+1v = -7 + 1v=−7+1

STEP 8

Calculate the value of vvv for the second case.v=−6v = -6v=−6

STEP 9

The solutions for the first equation are v=8v = 8v=8 and v=−6v = -6v=−6.

STEP 10

Solve the second equation ∣y4∣=2\left|\frac{y}{4}\right|=2∣∣​4y​∣∣​=2.Again, we have two cases for y4\frac{y}{4}4y​:1. y4=2\frac{y}{4} = 24y​=22. y4=−2\frac{y}{4} = -24y​=−2

STEP 11

Solve for yyy in the first case of the second equation.y4=2\frac{y}{4} = 24y​=2

STEP 12

Multiply both sides of the equation by 4.y=2×4y = 2 \times 4y=2×4

STEP 13

Calculate the value of yyy for the first case.y=8y = 8y=8

STEP 14

Solve for yyy in the second case of the second equation.y4=−2\frac{y}{4} = -24y​=−2

STEP 15

Multiply both sides of the equation by 4.y=−2×4y = -2 \times 4y=−2×4

STEP 16

Calculate the value of yyy for the second case.y=−8y = -8y=−8

STEP 17

The solutions for the second equation are y=8y = 8y=8 and y=−8y = -8y=−8.

STEP 18

Solve the third equation ∣−10k∣=30|-10 k|=30∣−10k∣=30.Again, we have two cases for −10k-10k−10k:1. −10k=30-10k = 30−10k=302. −10k=−30-10k = -30−10k=−30

STEP 19

Solve for kkk in the first case of the third equation.−10k=30-10k = 30−10k=30

STEP 20

Divide both sides of the equation by -10.k=30−10k = \frac{30}{-10}k=−1030​

STEP 21

Calculate the value of kkk for the first case.k=−3k = -3k=−3

STEP 22

Solve for kkk in the second case of the third equation.−10k=−30-10k = -30−10k=−30

STEP 23

Divide both sides of the equation by -10.k=−30−10k = \frac{-30}{-10}k=−10−30​

STEP 24

Calculate the value of kkk for the second case.k=3k = 3k=3

STEP 25

The solutions for the third equation are k=−3k = -3k=−3 and k=3k = 3k=3.

STEP 26

Solve the fourth equation ∣n+8∣=1|n+8|=1∣n+8∣=1.Again, we have two cases for n+8n+8n+8:1. n+8=1n+8 = 1n+8=12. n+8=−1n+8 = -1n+8=−1

STEP 27

Solve for nnn in the first case of the fourth equation.n+8=1n + 8 = 1n+8=1

STEP 28

Subtract 8 from both sides of the equation.n=1−8n = 1 - 8n=1−8

STEP 29

Calculate the value of nnn for the first case.n=−7n = -7n=−7

STEP 30

Solve for nnn in the second case of the fourth equation.n+8=−1n + 8 = -1n+8=−1

STEP 31

Subtract 8 from both sides of the equation.n=−1−8n = -1 - 8n=−1−8

STEP 32

Calculate the value of nnn for the second case.n=−9n = -9n=−9The solutions for the equations are: 1) v=8v = 8v=8 and v=−6v = -6v=−6 2) y=8y = 8y=8 and y=−8y = -8y=−8 3) k=−3k = -3k=−3 and k=3k = 3k=3 4) n=−7n = -7n=−7 and n=−9n = -9n=−9

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Solve the absolute value equations: $|v-1|=7$ , $\left|\frac{y}{4}\right|=2$ , $|-10 k|=30$ , $|n+8|=1$ .

Solve the first equation $|v-1|=7$ .
An absolute value equation $|x| = a$ has two solutions: $x = a$ and $x = -a$ . Therefore, we have two cases for $v-1$ :
1. $v-1 = 7$
2. $v-1 = -7$

Solve for $v$ in the first case of the first equation.
$v - 1 = 7$

Add 1 to both sides of the equation.
$v = 7 + 1$

Calculate the value of $v$ for the first case.
$v = 8$

Solve for $v$ in the second case of the first equation.
$v - 1 = -7$

Add 1 to both sides of the equation.
$v = -7 + 1$

Calculate the value of $v$ for the second case.
$v = -6$

The solutions for the first equation are $v = 8$ and $v = -6$ .

Solve the second equation $\left|\frac{y}{4}\right|=2$ .
Again, we have two cases for $\frac{y}{4}$ :
1. $\frac{y}{4} = 2$
2. $\frac{y}{4} = -2$

Solve for $y$ in the first case of the second equation.
$\frac{y}{4} = 2$

Multiply both sides of the equation by 4.
$y = 2 \times 4$

Calculate the value of $y$ for the first case.
$y = 8$

Solve for $y$ in the second case of the second equation.
$\frac{y}{4} = -2$

Multiply both sides of the equation by 4.
$y = -2 \times 4$

Calculate the value of $y$ for the second case.
$y = -8$

The solutions for the second equation are $y = 8$ and $y = -8$ .

Solve the third equation $|-10 k|=30$ .
Again, we have two cases for $-10k$ :
1. $-10k = 30$
2. $-10k = -30$

Solve for $k$ in the first case of the third equation.
$-10k = 30$

Divide both sides of the equation by -10.
$k = \frac{30}{-10}$

Calculate the value of $k$ for the first case.
$k = -3$

Solve for $k$ in the second case of the third equation.
$-10k = -30$

Divide both sides of the equation by -10.
$k = \frac{-30}{-10}$

Calculate the value of $k$ for the second case.
$k = 3$

The solutions for the third equation are $k = -3$ and $k = 3$ .

Solve the fourth equation $|n+8|=1$ .
Again, we have two cases for $n+8$ :
1. $n+8 = 1$
2. $n+8 = -1$

Solve for $n$ in the first case of the fourth equation.
$n + 8 = 1$

Subtract 8 from both sides of the equation.
$n = 1 - 8$

Calculate the value of $n$ for the first case.
$n = -7$

Solve for $n$ in the second case of the fourth equation.
$n + 8 = -1$

Subtract 8 from both sides of the equation.
$n = -1 - 8$

Calculate the value of $n$ for the second case.
$n = -9$
The solutions for the equations are: 1) $v = 8$ and $v = -6$ 2) $y = 8$ and $y = -8$ 3) $k = -3$ and $k = 3$ 4) $n = -7$ and $n = -9$