Finite Math Examples

$| 4 x + 7 | = x + 1$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 x + 7 = \pm (x + 1)$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$4 x + 7 = x + 1$

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$4 x + 7 - x = 1$

Step 2.2.2

Subtract $x$ from $4 x$ .

$3 x + 7 = 1$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Subtract $7$ from both sides of the equation.

$3 x = 1 - 7$

Step 2.3.2

Subtract $7$ from $1$ .

$3 x = - 6$

Step 2.4

Divide each term in $3 x = - 6$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 x = - 6$ by $3$ .

$\frac{3 x}{3} = \frac{- 6}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 6}{3}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 6$ by $3$ .

$x = - 2$

Step 2.5

Next, use the negative value of the $\pm$ to find the second solution.

$4 x + 7 = - (x + 1)$

Step 2.6

Simplify $- (x + 1)$ .

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Step 2.6.1

Rewrite.

$4 x + 7 = 0 + 0 - (x + 1)$

Step 2.6.2

Simplify by adding zeros.

$4 x + 7 = - (x + 1)$

Step 2.6.3

Apply the distributive property.

$4 x + 7 = - x - 1 \cdot 1$

Step 2.6.4

Multiply $- 1$ by $1$ .

$4 x + 7 = - x - 1$

Step 2.7

Move all terms containing $x$ to the left side of the equation.

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Step 2.7.1

Add $x$ to both sides of the equation.

$4 x + 7 + x = - 1$

Step 2.7.2

Add $4 x$ and $x$ .

$5 x + 7 = - 1$

Step 2.8

Move all terms not containing $x$ to the right side of the equation.

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Step 2.8.1

Subtract $7$ from both sides of the equation.

$5 x = - 1 - 7$

Step 2.8.2

Subtract $7$ from $- 1$ .

$5 x = - 8$

Step 2.9

Divide each term in $5 x = - 8$ by $5$ and simplify.

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Step 2.9.1

Divide each term in $5 x = - 8$ by $5$ .

$\frac{5 x}{5} = \frac{- 8}{5}$

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $5$ .

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Step 2.9.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 8}{5}$

Step 2.9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{5}$

Step 2.9.3

Simplify the right side.

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Step 2.9.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{5}$

Step 2.10

The complete solution is the result of both the positive and negative portions of the solution.

$x = - 2, - \frac{8}{5}$

Step 3

Exclude the solutions that do not make $| 4 x + 7 | = x + 1$ true.

$No solution$