Algebra Examples

$g (x) = 2 | 7 - x |$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = 2 | 7 - x |$ is $(7, 0)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $7 - x$ equal to $0$ . In this case, $7 - x = 0$ .

$7 - x = 0$

Step 1.2

Solve the equation $7 - x = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Subtract $7$ from both sides of the equation.

$- x = - 7$

Step 1.2.2

Divide each term in $- x = - 7$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 7$ by $- 1$ .

$x = 7$

Step 1.3

Replace the variable $x$ with $7$ in the expression.

$y = 2 | 7 - (7) |$

Step 1.4

Simplify $2 | 7 - (7) |$ .

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Step 1.4.1

Multiply $- 1$ by $7$ .

$y = 2 | 7 - 7 |$

Step 1.4.2

Subtract $7$ from $7$ .

$y = 2 | 0 |$

Step 1.4.3

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

$y = 2 \cdot 0$

Step 1.4.4

Multiply $2$ by $0$ .

$y = 0$

Step 1.5

The absolute value vertex is $(7, 0)$ .

$(7, 0)$

Step 2

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 3

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 3.1

Substitute the $x$ value $5$ into $f (x) = 2 | 7 - x |$ . In this case, the point is $(5, 4)$ .

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Step 3.1.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = 2 | 7 - (5) |$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply $- 1$ by $5$ .

$f (5) = 2 | 7 - 5 |$

Step 3.1.2.2

Subtract $5$ from $7$ .

$f (5) = 2 | 2 |$

Step 3.1.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$f (5) = 2 \cdot 2$

Step 3.1.2.4

Multiply $2$ by $2$ .

$f (5) = 4$

Step 3.1.2.5

The final answer is $4$ .

$y = 4$

Step 3.2

Substitute the $x$ value $6$ into $f (x) = 2 | 7 - x |$ . In this case, the point is $(6, 2)$ .

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Step 3.2.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 2 | 7 - (6) |$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Multiply $- 1$ by $6$ .

$f (6) = 2 | 7 - 6 |$

Step 3.2.2.2

Subtract $6$ from $7$ .

$f (6) = 2 | 1 |$

Step 3.2.2.3

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

$f (6) = 2 \cdot 1$

Step 3.2.2.4

Multiply $2$ by $1$ .

$f (6) = 2$

Step 3.2.2.5

The final answer is $2$ .

$y = 2$

Step 3.3

Substitute the $x$ value $9$ into $f (x) = 2 | 7 - x |$ . In this case, the point is $(9, 4)$ .

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Step 3.3.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = 2 | 7 - (9) |$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply $- 1$ by $9$ .

$f (9) = 2 | 7 - 9 |$

Step 3.3.2.2

Subtract $9$ from $7$ .

$f (9) = 2 | - 2 |$

Step 3.3.2.3

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$f (9) = 2 \cdot 2$

Step 3.3.2.4

Multiply $2$ by $2$ .

$f (9) = 4$

Step 3.3.2.5

The final answer is $4$ .

$y = 4$

Step 3.4

The absolute value can be graphed using the points around the vertex $(7, 0), (5, 4), (6, 2), (8, 2), (9, 4)$

$\begin{array}{c} x & y \\ 5 & 4 \\ 6 & 2 \\ 7 & 0 \\ 8 & 2 \\ 9 & 4 \end{array}$

Step 4