Algebra Examples

$f (x) = | x + 2 | - 3$

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 $x$ .

$0 = | x + 2 | - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $| x + 2 | - 3 = 0$ .

$| x + 2 | - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$| x + 2 | = 3$

Step 1.2.3

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

$x + 2 = \pm 3$

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.

$x + 2 = 3$

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

Subtract $2$ from both sides of the equation.

$x = 3 - 2$

Step 1.2.4.2.2

Subtract $2$ from $3$ .

$x = 1$

Step 1.2.4.3

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

$x + 2 = - 3$

Step 1.2.4.4

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

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Step 1.2.4.4.1

Subtract $2$ from both sides of the equation.

$x = - 3 - 2$

Step 1.2.4.4.2

Subtract $2$ from $- 3$ .

$x = - 5$

Step 1.2.4.5

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

$x = 1, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- 5, 0)$

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 $y$ .

$y = | (0) + 2 | - 3$

Step 2.2

Simplify $| (0) + 2 | - 3$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $2$ .

$y = | 2 | - 3$

Step 2.2.1.2

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

$y = 2 - 3$

Step 2.2.2

Subtract $3$ from $2$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (- 5, 0)$

y-intercept(s): $(0, - 1)$

Step 4