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Solution - Absolute value equations

Exact form:

a = 2, - \frac{1}{3}

a=2 , -\frac{1}{3}

Decimal form:

a = 2, - 0.333

a=2 , -0.333

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
$| 5 a - 3 | = | a + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 a - 3 \| = \| a + 5 \|$
$x = + y$	$(5 a - 3) = (a + 5)$
$x = - y$	$(5 a - 3) = - (a + 5)$
$+ x = y$	$(5 a - 3) = (a + 5)$
$- x = y$	$- (5 a - 3) = (a + 5)$

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:

$\| x \| = \| y \|$	$\| 5 a - 3 \| = \| a + 5 \|$
$x = + y$ , $+ x = y$	$(5 a - 3) = (a + 5)$
$x = - y$ , $- x = y$	$(5 a - 3) = - (a + 5)$

2. Solve the two equations for a

11 additional steps

$(5 a - 3) = (a + 5)$

Subtract from both sides:

$(5 a - 3) - a = (a + 5) - a$

Group like terms:

$(5 a - a) - 3 = (a + 5) - a$

Simplify the arithmetic:

$4 a - 3 = (a + 5) - a$

Group like terms:

$4 a - 3 = (a - a) + 5$

Simplify the arithmetic:

$4 a - 3 = 5$

Add to both sides:

$(4 a - 3) + 3 = 5 + 3$

Simplify the arithmetic:

$4 a = 5 + 3$

Simplify the arithmetic:

$4 a = 8$

Divide both sides by :

$\frac{(4 a)}{4} = \frac{8}{4}$

Simplify the fraction:

$a = \frac{8}{4}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(2 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$a = 2$

12 additional steps

$(5 a - 3) = - (a + 5)$

Expand the parentheses:

$(5 a - 3) = - a - 5$

Add to both sides:

$(5 a - 3) + a = (- a - 5) + a$

Group like terms:

$(5 a + a) - 3 = (- a - 5) + a$

Simplify the arithmetic:

$6 a - 3 = (- a - 5) + a$

Group like terms:

$6 a - 3 = (- a + a) - 5$

Simplify the arithmetic:

$6 a - 3 = - 5$

Add to both sides:

$(6 a - 3) + 3 = - 5 + 3$

Simplify the arithmetic:

$6 a = - 5 + 3$

Simplify the arithmetic:

$6 a = - 2$

Divide both sides by :

$\frac{(6 a)}{6} = \frac{- 2}{6}$

Simplify the fraction:

$a = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{- 1}{3}$

3. List the solutions

$a = 2, - \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 a - 3 |$
$y = | a + 5 |$
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 a

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 a

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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