Question

Solve the absolute value equation −3−|6−3x|+7=4 .(1 point)
Responses

x=−3
x equals negative 3

no solution
no solution

x=2
x equals 2

x=0

Answer 1

Solve the absolute value equation 4|7−x|+1=−1 .(1 point)
Responses

no solution
no solution

x=132 and x=152
x equals Start Fraction 13 over 2 End Fraction and x equals Start Fraction 15 over 2 End Fraction

x=152
x equals Start Fraction 15 over 2 End Fraction

x=7
x equals 7
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page 16 of 16

Answer 2

Solve the absolute value equation |12−3x|=|9+4x|.(1 point)
Responses

x=−21 and x=37
x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction

no solution
no solution

x=4
x equals 4

x=37 and x=3
x equals Start Fraction 3 over 7 End Fraction and x equals 3
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page 16 of 16

Answer 3

To solve the equation \(-3 - |6 - 3x| + 7 = 4\), we first simplify the left-hand side:

\[
-3 + 7 - |6 - 3x| = 4
\]

This simplifies to:

\[
4 - |6 - 3x| = 4
\]

Now, subtract 4 from both sides:

\[
- |6 - 3x| = 0
\]

This implies:

\[
|6 - 3x| = 0
\]

The absolute value of a number is zero only if that number itself is zero, so we set up the equation:

\[
6 - 3x = 0
\]

Now, solve for \(x\):

\[
3x = 6
\]
\[
x = 2
\]

Now, we'll check if this value satisfies the original equation:

1. Substitute \(x = 2\) into the expression \(6 - 3x\):

\[
6 - 3(2) = 6 - 6 = 0
\]

2. Evaluate the left-hand side of the original equation:

\[
-3 - |0| + 7 = -3 - 0 + 7 = 4
\]

Since both sides of the original equation are equal, \(x = 2\) is indeed a solution.

Thus, the solution to the equation is:

\[
\boxed{x=2}
\]

Answer 4

To solve the equation \( 4|7 - x| + 1 = -1 \), we can start by moving \(1\) to the right side:

\[
4|7 - x| + 1 - 1 = -1 - 1
\]

This simplifies to:

\[
4|7 - x| = -2
\]

Next, divide both sides by \(4\):

\[
|7 - x| = -\frac{1}{2}
\]

However, the absolute value of any expression is always non-negative (i.e., it is always \( \geq 0 \)). Therefore, there is no possible \(x\) such that \( |7 - x| = -\frac{1}{2} \).

Thus, the equation has **no solution**.

The final answer is:

\[
\boxed{\text{no solution}}
\]

Answer 5

To solve the equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases for the absolute value expressions.

### Case 1: \(12 - 3x = 9 + 4x\)

1. Rearranging the equation:
\[
12 - 9 = 4x + 3x
\]
\[
3 = 7x
\]
\[
x = \frac{3}{7}
\]

### Case 2: \(12 - 3x = -(9 + 4x)\)

1. Rearranging gives:
\[
12 - 3x = -9 - 4x
\]
\[
12 + 9 = -4x + 3x
\]
\[
21 = -x
\]
\[
x = -21
\]

### Case 3: \(-(12 - 3x) = 9 + 4x\)

1. Rearranging gives:
\[
-12 + 3x = 9 + 4x
\]
\[
3x - 4x = 9 + 12
\]
\[
-x = 21
\]
\[
x = -21
\]

### Case 4: \(-(12 - 3x) = -(9 + 4x)\)

1. Rearranging gives:
\[
-12 + 3x = -9 - 4x
\]
\[
3x + 4x = -9 + 12
\]
\[
7x = 3
\]
\[
x = \frac{3}{7}
\]

### Summary of Solutions

From the four cases, we have found two unique solutions:

1. \( x = \frac{3}{7} \)
2. \( x = -21 \)

Thus, the solutions to the equation \( |12 - 3x| = |9 + 4x| \) are:

\[
\boxed{x = -21 \text{ and } x = \frac{3}{7}}
\]