Algebra Examples

$y = | 1 - \frac{3 x}{4} | - 7$

Step 1

Reorder $1$ and $- \frac{3 x}{4}$ .

$y = | - \frac{3 x}{4} + 1 | - 7$

Step 2

Find the absolute value vertex. In this case, the vertex for $y = | - \frac{3 x}{4} + 1 | - 7$ is $(\frac{4}{3}, - 7)$ .

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Step 2.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $- \frac{3 x}{4} + 1$ equal to $0$ . In this case, $- \frac{3 x}{4} + 1 = 0$ .

$- \frac{3 x}{4} + 1 = 0$

Step 2.2

Solve the equation $- \frac{3 x}{4} + 1 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$- \frac{3 x}{4} = - 1$

Step 2.2.2

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3 x}{4}) = - \frac{4}{3} \cdot - 1$

Step 2.2.3

Simplify both sides of the equation.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Simplify $- \frac{4}{3} (- \frac{3 x}{4})$ .

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Step 2.2.3.1.1.1

Cancel the common factor of $4$ .

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Step 2.2.3.1.1.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{3 x}{4}) = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.1.2

Move the leading negative in $- \frac{3 x}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 3 x}{4} = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.3.1.1.2.1

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.2.3

Rewrite the expression.

$- - x = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.3

Multiply.

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Step 2.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{4}{3} \cdot - 1$

Step 2.2.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{4}{3} \cdot - 1$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply $- \frac{4}{3} \cdot - 1$ .

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Step 2.2.3.2.1.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{4}{3})$

Step 2.2.3.2.1.2

Multiply $\frac{4}{3}$ by $1$ .

$x = \frac{4}{3}$

Step 2.3

Replace the variable $x$ with $\frac{4}{3}$ in the expression.

$y = | - \frac{3 (\frac{4}{3})}{4} + 1 | - 7$

Step 2.4

Simplify $| - \frac{3 (\frac{4}{3})}{4} + 1 | - 7$ .

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Combine $3$ and $\frac{4}{3}$ .

$y = | - \frac{\frac{3 \cdot 4}{3}}{4} + 1 | - 7$

Step 2.4.1.1.2

Multiply $3$ by $4$ .

$y = | - \frac{\frac{12}{3}}{4} + 1 | - 7$

Step 2.4.1.1.3

Divide $12$ by $3$ .

$y = | - \frac{4}{4} + 1 | - 7$

Step 2.4.1.1.4

Cancel the common factor of $4$ .

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Step 2.4.1.1.4.1

Cancel the common factor.

$y = | - \frac{4}{4} + 1 | - 7$

Step 2.4.1.1.4.2

Rewrite the expression.

$y = | - 1 \cdot 1 + 1 | - 7$

Step 2.4.1.1.5

Multiply $- 1$ by $1$ .

$y = | - 1 + 1 | - 7$

Step 2.4.1.2

Add $- 1$ and $1$ .

$y = | 0 | - 7$

Step 2.4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$y = 0 - 7$

Step 2.4.2

Subtract $7$ from $0$ .

$y = - 7$

Step 2.5

The absolute value vertex is $(\frac{4}{3}, - 7)$ .

$(\frac{4}{3}, - 7)$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

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Step 4.1

Substitute the $x$ value $- 1$ into $f (x) = | - \frac{3 x}{4} + 1 | - 7$ . In this case, the point is $(- 1, - \frac{21}{4})$ .

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Step 4.1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = | - \frac{3 (- 1)}{4} + 1 | - 7$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Simplify each term.

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Step 4.1.2.1.1.1

Multiply $3$ by $- 1$ .

$f (- 1) = | - \frac{- 3}{4} + 1 | - 7$

Step 4.1.2.1.1.2

Move the negative in front of the fraction.

$f (- 1) = | \frac{3}{4} + 1 | - 7$

Step 4.1.2.1.1.3

Multiply $- - \frac{3}{4}$ .

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Step 4.1.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = | 1 (\frac{3}{4}) + 1 | - 7$

Step 4.1.2.1.1.3.2

Multiply $\frac{3}{4}$ by $1$ .

$f (- 1) = | \frac{3}{4} + 1 | - 7$

Step 4.1.2.1.2

Write $1$ as a fraction with a common denominator.

$f (- 1) = | \frac{3}{4} + \frac{4}{4} | - 7$

Step 4.1.2.1.3

Combine the numerators over the common denominator.

$f (- 1) = | \frac{3 + 4}{4} | - 7$

Step 4.1.2.1.4

Add $3$ and $4$ .

$f (- 1) = | \frac{7}{4} | - 7$

Step 4.1.2.1.5

$\frac{7}{4}$ is approximately $1.75$ which is positive so remove the absolute value

$f (- 1) = \frac{7}{4} - 7$

Step 4.1.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 1) = \frac{7}{4} - 7 \cdot \frac{4}{4}$

Step 4.1.2.3

Combine $- 7$ and $\frac{4}{4}$ .

$f (- 1) = \frac{7}{4} + \frac{- 7 \cdot 4}{4}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{7 - 7 \cdot 4}{4}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $- 7$ by $4$ .

$f (- 1) = \frac{7 - 28}{4}$

Step 4.1.2.5.2

Subtract $28$ from $7$ .

$f (- 1) = \frac{- 21}{4}$

Step 4.1.2.6

Move the negative in front of the fraction.

$f (- 1) = - \frac{21}{4}$

Step 4.1.2.7

The final answer is $- \frac{21}{4}$ .

$y = - \frac{21}{4}$

Step 4.2

Substitute the $x$ value $0$ into $f (x) = | - \frac{3 x}{4} + 1 | - 7$ . In this case, the point is $(0, - 6)$ .

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Step 4.2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = | - \frac{3 (0)}{4} + 1 | - 7$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Simplify each term.

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Step 4.2.2.1.1.1

Cancel the common factor of $0$ and $4$ .

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Step 4.2.2.1.1.1.1

Factor $4$ out of $3 (0)$ .

$f (0) = | - \frac{4 (3 \cdot (0))}{4} + 1 | - 7$

Step 4.2.2.1.1.1.2

Cancel the common factors.

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Step 4.2.2.1.1.1.2.1

Factor $4$ out of $4$ .

$f (0) = | - \frac{4 (3 \cdot (0))}{4 (1)} + 1 | - 7$

Step 4.2.2.1.1.1.2.2

Cancel the common factor.

$f (0) = | - \frac{4 (3 \cdot (0))}{4 \cdot 1} + 1 | - 7$

Step 4.2.2.1.1.1.2.3

Rewrite the expression.

$f (0) = | - \frac{3 \cdot (0)}{1} + 1 | - 7$

Step 4.2.2.1.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$f (0) = | - (3 \cdot (0)) + 1 | - 7$

Step 4.2.2.1.1.2

Multiply $- (3 \cdot (0))$ .

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Step 4.2.2.1.1.2.1

Multiply $3$ by $0$ .

$f (0) = | - 0 + 1 | - 7$

Step 4.2.2.1.1.2.2

Multiply $- 1$ by $0$ .

$f (0) = | 0 + 1 | - 7$

Step 4.2.2.1.2

Add $0$ and $1$ .

$f (0) = | 1 | - 7$

Step 4.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$f (0) = 1 - 7$

Step 4.2.2.2

Subtract $7$ from $1$ .

$f (0) = - 6$

Step 4.2.2.3

The final answer is $- 6$ .

$y = - 6$

Step 4.3

Substitute the $x$ value $1$ into $f (x) = | - \frac{3 x}{4} + 1 | - 7$ . In this case, the point is $(1, - \frac{27}{4})$ .

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Step 4.3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = | - \frac{3 (1)}{4} + 1 | - 7$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $3$ by $1$ .

$f (1) = | - \frac{3}{4} + 1 | - 7$

Step 4.3.2.1.2

Write $1$ as a fraction with a common denominator.

$f (1) = | - \frac{3}{4} + \frac{4}{4} | - 7$

Step 4.3.2.1.3

Combine the numerators over the common denominator.

$f (1) = | \frac{- 3 + 4}{4} | - 7$

Step 4.3.2.1.4

Add $- 3$ and $4$ .

$f (1) = | \frac{1}{4} | - 7$

Step 4.3.2.1.5

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$f (1) = \frac{1}{4} - 7$

Step 4.3.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (1) = \frac{1}{4} - 7 \cdot \frac{4}{4}$

Step 4.3.2.3

Combine $- 7$ and $\frac{4}{4}$ .

$f (1) = \frac{1}{4} + \frac{- 7 \cdot 4}{4}$

Step 4.3.2.4

Combine the numerators over the common denominator.

$f (1) = \frac{1 - 7 \cdot 4}{4}$

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Multiply $- 7$ by $4$ .

$f (1) = \frac{1 - 28}{4}$

Step 4.3.2.5.2

Subtract $28$ from $1$ .

$f (1) = \frac{- 27}{4}$

Step 4.3.2.6

Move the negative in front of the fraction.

$f (1) = - \frac{27}{4}$

Step 4.3.2.7

The final answer is $- \frac{27}{4}$ .

$y = - \frac{27}{4}$

Step 4.4

Substitute the $x$ value $2$ into $f (x) = | - \frac{3 x}{4} + 1 | - 7$ . In this case, the point is $(2, - \frac{13}{2})$ .

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Step 4.4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = | - \frac{3 (2)}{4} + 1 | - 7$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Cancel the common factor of $2$ and $4$ .

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Step 4.4.2.1.1.1

Factor $2$ out of $3 (2)$ .

$f (2) = | - \frac{2 \cdot 3}{4} + 1 | - 7$

Step 4.4.2.1.1.2

Cancel the common factors.

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Step 4.4.2.1.1.2.1

Factor $2$ out of $4$ .

$f (2) = | - \frac{2 \cdot 3}{2 \cdot 2} + 1 | - 7$

Step 4.4.2.1.1.2.2

Cancel the common factor.

$f (2) = | - \frac{2 \cdot 3}{2 \cdot 2} + 1 | - 7$

Step 4.4.2.1.1.2.3

Rewrite the expression.

$f (2) = | - \frac{3}{2} + 1 | - 7$

Step 4.4.2.1.2

Write $1$ as a fraction with a common denominator.

$f (2) = | - \frac{3}{2} + \frac{2}{2} | - 7$

Step 4.4.2.1.3

Combine the numerators over the common denominator.

$f (2) = | \frac{- 3 + 2}{2} | - 7$

Step 4.4.2.1.4

Add $- 3$ and $2$ .

$f (2) = | \frac{- 1}{2} | - 7$

Step 4.4.2.1.5

Move the negative in front of the fraction.

$f (2) = | - \frac{1}{2} | - 7$

Step 4.4.2.1.6

$- \frac{1}{2}$ is approximately $- 0.5$ which is negative so negate $- \frac{1}{2}$ and remove the absolute value

$f (2) = \frac{1}{2} - 7$

Step 4.4.2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = \frac{1}{2} - 7 \cdot \frac{2}{2}$

Step 4.4.2.3

Combine $- 7$ and $\frac{2}{2}$ .

$f (2) = \frac{1}{2} + \frac{- 7 \cdot 2}{2}$

Step 4.4.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{1 - 7 \cdot 2}{2}$

Step 4.4.2.5

Simplify the numerator.

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Step 4.4.2.5.1

Multiply $- 7$ by $2$ .

$f (2) = \frac{1 - 14}{2}$

Step 4.4.2.5.2

Subtract $14$ from $1$ .

$f (2) = \frac{- 13}{2}$

Step 4.4.2.6

Move the negative in front of the fraction.

$f (2) = - \frac{13}{2}$

Step 4.4.2.7

The final answer is $- \frac{13}{2}$ .

$y = - \frac{13}{2}$

Step 4.5

The absolute value can be graphed using the points around the vertex $(\frac{4}{3}, - 7), (- 1, - 5.25), (0, - 6), (1, - 6.75), (2, - 6.5)$

$\begin{array}{c} x & y \\ - 1 & - 5.25 \\ 0 & - 6 \\ 1 & - 6.75 \\ \frac{4}{3} & - 7 \\ 2 & - 6.5 \end{array}$

Step 5