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Solution - Absolute value equations

Exact form:

a = \frac{2}{5}, - 2

a=\frac{2}{5} , -2

Decimal form:

a = 0.4, - 2

a=0.4 , -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
$| 5 a - 2 | = | - 5 a + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 a - 2 \| = \| - 5 a + 2 \|$
$x = + y$	$(5 a - 2) = (- 5 a + 2)$
$x = - y$	$(5 a - 2) = - (- 5 a + 2)$
$+ x = y$	$(5 a - 2) = (- 5 a + 2)$
$- x = y$	$- (5 a - 2) = (- 5 a + 2)$

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:

$\| x \| = \| y \|$	$\| 5 a - 2 \| = \| - 5 a + 2 \|$
$x = + y$ , $+ x = y$	$(5 a - 2) = (- 5 a + 2)$
$x = - y$ , $- x = y$	$(5 a - 2) = - (- 5 a + 2)$

2. Solve the two equations for a

11 additional steps

$(5 a - 2) = (- 5 a + 2)$

Add to both sides:

$(5 a - 2) + 5 a = (- 5 a + 2) + 5 a$

Group like terms:

$(5 a + 5 a) - 2 = (- 5 a + 2) + 5 a$

Simplify the arithmetic:

$10 a - 2 = (- 5 a + 2) + 5 a$

Group like terms:

$10 a - 2 = (- 5 a + 5 a) + 2$

Simplify the arithmetic:

$10 a - 2 = 2$

Add to both sides:

$(10 a - 2) + 2 = 2 + 2$

Simplify the arithmetic:

$10 a = 2 + 2$

Simplify the arithmetic:

$10 a = 4$

Divide both sides by :

$\frac{(10 a)}{10} = \frac{4}{10}$

Simplify the fraction:

$a = \frac{4}{10}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(2 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{2}{5}$

5 additional steps

$(5 a - 2) = - (- 5 a + 2)$

Expand the parentheses:

$(5 a - 2) = 5 a - 2$

Subtract from both sides:

$(5 a - 2) - 5 a = (5 a - 2) - 5 a$

Group like terms:

$(5 a - 5 a) - 2 = (5 a - 2) - 5 a$

Simplify the arithmetic:

$- 2 = (5 a - 2) - 5 a$

Group like terms:

$- 2 = (5 a - 5 a) - 2$

Simplify the arithmetic:

$- 2 = - 2$

3. List the solutions

$a = \frac{2}{5}, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 a - 2 |$
$y = | - 5 a + 2 |$
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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