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Solution - Absolute value equations

Exact form:

j = - 2

j=-2

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
$| j - 5 | = | j + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| j - 5 \| = \| j + 9 \|$
$x = + y$	$(j - 5) = (j + 9)$
$x = - y$	$(j - 5) = - (j + 9)$
$+ x = y$	$(j - 5) = (j + 9)$
$- x = y$	$- (j - 5) = (j + 9)$

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:

$\| x \| = \| y \|$	$\| j - 5 \| = \| j + 9 \|$
$x = + y$ , $+ x = y$	$(j - 5) = (j + 9)$
$x = - y$ , $- x = y$	$(j - 5) = - (j + 9)$

2. Solve the two equations for j

5 additional steps

$(j - 5) = (j + 9)$

Subtract from both sides:

$(j - 5) - j = (j + 9) - j$

Group like terms:

$(j - j) - 5 = (j + 9) - j$

Simplify the arithmetic:

$- 5 = (j + 9) - j$

Group like terms:

$- 5 = (j - j) + 9$

Simplify the arithmetic:

$- 5 = 9$

The statement is false:

$- 5 = 9$

The equation is false so it has no solution.

12 additional steps

$(j - 5) = - (j + 9)$

Expand the parentheses:

$(j - 5) = - j - 9$

Add to both sides:

$(j - 5) + j = (- j - 9) + j$

Group like terms:

$(j + j) - 5 = (- j - 9) + j$

Simplify the arithmetic:

$2 j - 5 = (- j - 9) + j$

Group like terms:

$2 j - 5 = (- j + j) - 9$

Simplify the arithmetic:

$2 j - 5 = - 9$

Add to both sides:

$(2 j - 5) + 5 = - 9 + 5$

Simplify the arithmetic:

$2 j = - 9 + 5$

Simplify the arithmetic:

$2 j = - 4$

Divide both sides by :

$\frac{(2 j)}{2} = \frac{- 4}{2}$

Simplify the fraction:

$j = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

$j = \frac{(- 2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$j = - 2$

3. Graph

Each line represents the function of one side of the equation:
$y = | j - 5 |$
$y = | j + 9 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved