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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

5 answers

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 \(6 - 3x\):

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

  1. 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:

\[ \boxed{x=2} \]

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

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:

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

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

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:

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