Algebra Examples

$g (x) = | \frac{3}{7} (x - 4) | + 2$

Step 1

Find the absolute value vertex. In this case, the vertex for $y = | \frac{3 x}{7} - \frac{12}{7} | + 2$ is $(4, 2)$ .

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Step 1.1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\frac{3 x}{7} - \frac{12}{7}$ equal to $0$ . In this case, $\frac{3 x}{7} - \frac{12}{7} = 0$ .

$\frac{3 x}{7} - \frac{12}{7} = 0$

Step 1.2

Solve the equation $\frac{3 x}{7} - \frac{12}{7} = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 1.2.1

Add $\frac{12}{7}$ to both sides of the equation.

$\frac{3 x}{7} = \frac{12}{7}$

Step 1.2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 x = 12$

Step 1.2.3

Divide each term in $3 x = 12$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 x = 12$ by $3$ .

$\frac{3 x}{3} = \frac{12}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{12}{3}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $12$ by $3$ .

$x = 4$

Step 1.3

Replace the variable $x$ with $4$ in the expression.

$y = | \frac{3 (4)}{7} - \frac{12}{7} | + 2$

Step 1.4

Simplify $| \frac{3 (4)}{7} - \frac{12}{7} | + 2$ .

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Combine the numerators over the common denominator.

$y = | \frac{3 (4) - 12}{7} | + 2$

Step 1.4.1.2

Multiply $3$ by $4$ .

$y = | \frac{12 - 12}{7} | + 2$

Step 1.4.1.3

Subtract $12$ from $12$ .

$y = | \frac{0}{7} | + 2$

Step 1.4.1.4

Divide $0$ by $7$ .

$y = | 0 | + 2$

Step 1.4.1.5

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

$y = 0 + 2$

Step 1.4.2

Add $0$ and $2$ .

$y = 2$

Step 1.5

The absolute value vertex is $(4, 2)$ .

$(4, 2)$

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 $2$ into $f (x) = | \frac{3 x}{7} - \frac{12}{7} | + 2$ . In this case, the point is $(2, \frac{20}{7})$ .

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Step 3.1.1

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

$f (2) = | \frac{3 (2)}{7} - \frac{12}{7} | + 2$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Combine the numerators over the common denominator.

$f (2) = | \frac{3 (2) - 12}{7} | + 2$

Step 3.1.2.1.2

Multiply $3$ by $2$ .

$f (2) = | \frac{6 - 12}{7} | + 2$

Step 3.1.2.1.3

Subtract $12$ from $6$ .

$f (2) = | \frac{- 6}{7} | + 2$

Step 3.1.2.1.4

Move the negative in front of the fraction.

$f (2) = | - \frac{6}{7} | + 2$

Step 3.1.2.1.5

$- \frac{6}{7}$ is approximately $- 0.\overline{857142}$ which is negative so negate $- \frac{6}{7}$ and remove the absolute value

$f (2) = \frac{6}{7} + 2$

Step 3.1.2.2

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

$f (2) = \frac{6}{7} + 2 \cdot \frac{7}{7}$

Step 3.1.2.3

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

$f (2) = \frac{6}{7} + \frac{2 \cdot 7}{7}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (2) = \frac{6 + 2 \cdot 7}{7}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply $2$ by $7$ .

$f (2) = \frac{6 + 14}{7}$

Step 3.1.2.5.2

Add $6$ and $14$ .

$f (2) = \frac{20}{7}$

Step 3.1.2.6

The final answer is $\frac{20}{7}$ .

$y = \frac{20}{7}$

Step 3.2

Substitute the $x$ value $3$ into $f (x) = | \frac{3 x}{7} - \frac{12}{7} | + 2$ . In this case, the point is $(3, \frac{17}{7})$ .

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Step 3.2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = | \frac{3 (3)}{7} - \frac{12}{7} | + 2$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Combine the numerators over the common denominator.

$f (3) = | \frac{3 (3) - 12}{7} | + 2$

Step 3.2.2.1.2

Multiply $3$ by $3$ .

$f (3) = | \frac{9 - 12}{7} | + 2$

Step 3.2.2.1.3

Subtract $12$ from $9$ .

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

Step 3.2.2.1.4

Move the negative in front of the fraction.

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

Step 3.2.2.1.5

$- \frac{3}{7}$ is approximately $- 0.\overline{428571}$ which is negative so negate $- \frac{3}{7}$ and remove the absolute value

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

Step 3.2.2.2

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

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

Step 3.2.2.3

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

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

Step 3.2.2.4

Combine the numerators over the common denominator.

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

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Multiply $2$ by $7$ .

$f (3) = \frac{3 + 14}{7}$

Step 3.2.2.5.2

Add $3$ and $14$ .

$f (3) = \frac{17}{7}$

Step 3.2.2.6

The final answer is $\frac{17}{7}$ .

$y = \frac{17}{7}$

Step 3.3

Substitute the $x$ value $6$ into $f (x) = | \frac{3 x}{7} - \frac{12}{7} | + 2$ . In this case, the point is $(6, \frac{20}{7})$ .

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Step 3.3.1

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

$f (6) = | \frac{3 (6)}{7} - \frac{12}{7} | + 2$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Combine the numerators over the common denominator.

$f (6) = | \frac{3 (6) - 12}{7} | + 2$

Step 3.3.2.1.2

Multiply $3$ by $6$ .

$f (6) = | \frac{18 - 12}{7} | + 2$

Step 3.3.2.1.3

Subtract $12$ from $18$ .

$f (6) = | \frac{6}{7} | + 2$

Step 3.3.2.1.4

$\frac{6}{7}$ is approximately $0.\overline{857142}$ which is positive so remove the absolute value

$f (6) = \frac{6}{7} + 2$

Step 3.3.2.2

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

$f (6) = \frac{6}{7} + 2 \cdot \frac{7}{7}$

Step 3.3.2.3

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

$f (6) = \frac{6}{7} + \frac{2 \cdot 7}{7}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$f (6) = \frac{6 + 2 \cdot 7}{7}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $2$ by $7$ .

$f (6) = \frac{6 + 14}{7}$

Step 3.3.2.5.2

Add $6$ and $14$ .

$f (6) = \frac{20}{7}$

Step 3.3.2.6

The final answer is $\frac{20}{7}$ .

$y = \frac{20}{7}$

Step 3.4

The absolute value can be graphed using the points around the vertex $(4, 2), (2, 2.86), (3, 2.43), (5, 2.43), (6, 2.86)$

$\begin{array}{c} x & y \\ 2 & 2.86 \\ 3 & 2.43 \\ 4 & 2 \\ 5 & 2.43 \\ 6 & 2.86 \end{array}$

Step 4