Algebra Examples

$f (x) = | 2 x - 5 | - 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = | 2 x - 5 | - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $| 2 x - 5 | - 1 = 0$ .

$| 2 x - 5 | - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$| 2 x - 5 | = 1$

Step 1.2.3

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

$2 x - 5 = \pm 1$

Step 1.2.4

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

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Step 1.2.4.1

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

$2 x - 5 = 1$

Step 1.2.4.2

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

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Step 1.2.4.2.1

Add $5$ to both sides of the equation.

$2 x = 1 + 5$

Step 1.2.4.2.2

Add $1$ and $5$ .

$2 x = 6$

Step 1.2.4.3

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 1.2.4.3.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 1.2.4.3.2

Simplify the left side.

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Step 1.2.4.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 1.2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 1.2.4.3.3

Simplify the right side.

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Step 1.2.4.3.3.1

Divide $6$ by $2$ .

$x = 3$

Step 1.2.4.4

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

$2 x - 5 = - 1$

Step 1.2.4.5

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

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Step 1.2.4.5.1

Add $5$ to both sides of the equation.

$2 x = - 1 + 5$

Step 1.2.4.5.2

Add $- 1$ and $5$ .

$2 x = 4$

Step 1.2.4.6

Divide each term in $2 x = 4$ by $2$ and simplify.

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Step 1.2.4.6.1

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Step 1.2.4.6.2

Simplify the left side.

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Step 1.2.4.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Step 1.2.4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Step 1.2.4.6.3

Simplify the right side.

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Step 1.2.4.6.3.1

Divide $4$ by $2$ .

$x = 2$

Step 1.2.4.7

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

$x = 3, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = | 2 (0) - 5 | - 1$

Step 2.2

Simplify $| 2 (0) - 5 | - 1$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $0$ .

$y = | 0 - 5 | - 1$

Step 2.2.1.2

Subtract $5$ from $0$ .

$y = | - 5 | - 1$

Step 2.2.1.3

The absolute value is the distance between a number and zero. The distance between $- 5$ and $0$ is $5$ .

$y = 5 - 1$

Step 2.2.2

Subtract $1$ from $5$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (2, 0)$

y-intercept(s): $(0, 4)$

Step 4