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Solution - Absolute value equations

Exact form:

a = \frac{17}{6}, \frac{17}{4}

a=\frac{17}{6} , \frac{17}{4}

Mixed number form:

a = 2 \frac{5}{6}, 4 \frac{1}{4}

a=2\frac{5}{6} , 4\frac{1}{4}

Decimal form:

a = 2.833, 4.25

a=2.833 , 4.25

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
$| a | = | - 5 a + 17 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| a \| = \| - 5 a + 17 \|$
$x = + y$	$(a) = (- 5 a + 17)$
$x = - y$	$(a) = - (- 5 a + 17)$
$+ x = y$	$(a) = (- 5 a + 17)$
$- x = y$	$- (a) = (- 5 a + 17)$

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:

$\| x \| = \| y \|$	$\| a \| = \| - 5 a + 17 \|$
$x = + y$ , $+ x = y$	$(a) = (- 5 a + 17)$
$x = - y$ , $- x = y$	$(a) = - (- 5 a + 17)$

2. Solve the two equations for a

5 additional steps

$a = (- 5 a + 17)$

Add to both sides:

$a + 5 a = (- 5 a + 17) + 5 a$

Simplify the arithmetic:

$6 a = (- 5 a + 17) + 5 a$

Group like terms:

$6 a = (- 5 a + 5 a) + 17$

Simplify the arithmetic:

$6 a = 17$

Divide both sides by :

$\frac{(6 a)}{6} = \frac{17}{6}$

Simplify the fraction:

$a = \frac{17}{6}$

8 additional steps

$a = - (- 5 a + 17)$

Expand the parentheses:

$a = 5 a - 17$

Subtract from both sides:

$a - 5 a = (5 a - 17) - 5 a$

Simplify the arithmetic:

$- 4 a = (5 a - 17) - 5 a$

Group like terms:

$- 4 a = (5 a - 5 a) - 17$

Simplify the arithmetic:

$- 4 a = - 17$

Divide both sides by :

$\frac{(- 4 a)}{- 4} = \frac{- 17}{- 4}$

Cancel out the negatives:

$\frac{4 a}{4} = \frac{- 17}{- 4}$

Simplify the fraction:

$a = \frac{- 17}{- 4}$

Cancel out the negatives:

$a = \frac{17}{4}$

3. List the solutions

$a = \frac{17}{6}, \frac{17}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | a |$
$y = | - 5 a + 17 |$
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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