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Solution - Absolute value equations

Exact form:

j = - 3, - 1

j=-3 , -1

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| j | = | 2 j + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| j \| = \| 2 j + 3 \|$
$x = + y$	$(j) = (2 j + 3)$
$x = - y$	$(j) = - (2 j + 3)$
$+ x = y$	$(j) = (2 j + 3)$
$- x = y$	$- (j) = (2 j + 3)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| j \| = \| 2 j + 3 \|$
$x = + y$ , $+ x = y$	$(j) = (2 j + 3)$
$x = - y$ , $- x = y$	$(j) = - (2 j + 3)$

2. Solve the two equations for j

6 additional steps

$j = (2 j + 3)$

Subtract from both sides:

$j - 2 j = (2 j + 3) - 2 j$

Simplify the arithmetic:

$- j = (2 j + 3) - 2 j$

Group like terms:

$- j = (2 j - 2 j) + 3$

Simplify the arithmetic:

$- j = 3$

Multiply both sides by :

$- j \cdot - 1 = 3 \cdot - 1$

Remove the one(s):

$j = 3 \cdot - 1$

Simplify the arithmetic:

$j = - 3$

7 additional steps

$j = - (2 j + 3)$

Expand the parentheses:

$j = - 2 j - 3$

Add to both sides:

$j + 2 j = (- 2 j - 3) + 2 j$

Simplify the arithmetic:

$3 j = (- 2 j - 3) + 2 j$

Group like terms:

$3 j = (- 2 j + 2 j) - 3$

Simplify the arithmetic:

$3 j = - 3$

Divide both sides by :

$\frac{(3 j)}{3} = \frac{- 3}{3}$

Simplify the fraction:

$j = \frac{- 3}{3}$

Simplify the fraction:

$j = - 1$

3. List the solutions

$j = - 3, - 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | j |$
$y = | 2 j + 3 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for j

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for j

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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