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Solution - Absolute value equations

Exact form:

a = \frac{15}{2}, - \frac{5}{4}

a=\frac{15}{2} , -\frac{5}{4}

Mixed number form:

a = 7 \frac{1}{2}, - 1 \frac{1}{4}

a=7\frac{1}{2} , -1\frac{1}{4}

Decimal form:

a = 7.5, - 1.25

a=7.5 , -1.25

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
$| 3 a - 5 | = | a + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 a - 5 \| = \| a + 10 \|$
$x = + y$	$(3 a - 5) = (a + 10)$
$x = - y$	$(3 a - 5) = - (a + 10)$
$+ x = y$	$(3 a - 5) = (a + 10)$
$- x = y$	$- (3 a - 5) = (a + 10)$

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:

$\| x \| = \| y \|$	$\| 3 a - 5 \| = \| a + 10 \|$
$x = + y$ , $+ x = y$	$(3 a - 5) = (a + 10)$
$x = - y$ , $- x = y$	$(3 a - 5) = - (a + 10)$

2. Solve the two equations for a

9 additional steps

$(3 a - 5) = (a + 10)$

Subtract from both sides:

$(3 a - 5) - a = (a + 10) - a$

Group like terms:

$(3 a - a) - 5 = (a + 10) - a$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 a - 5 = 10$

Add to both sides:

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

Simplify the arithmetic:

$2 a = 10 + 5$

Simplify the arithmetic:

$2 a = 15$

Divide both sides by :

$\frac{(2 a)}{2} = \frac{15}{2}$

Simplify the fraction:

$a = \frac{15}{2}$

10 additional steps

$(3 a - 5) = - (a + 10)$

Expand the parentheses:

$(3 a - 5) = - a - 10$

Add to both sides:

$(3 a - 5) + a = (- a - 10) + a$

Group like terms:

$(3 a + a) - 5 = (- a - 10) + a$

Simplify the arithmetic:

$4 a - 5 = (- a - 10) + a$

Group like terms:

$4 a - 5 = (- a + a) - 10$

Simplify the arithmetic:

$4 a - 5 = - 10$

Add to both sides:

$(4 a - 5) + 5 = - 10 + 5$

Simplify the arithmetic:

$4 a = - 10 + 5$

Simplify the arithmetic:

$4 a = - 5$

Divide both sides by :

$\frac{(4 a)}{4} = \frac{- 5}{4}$

Simplify the fraction:

$a = \frac{- 5}{4}$

3. List the solutions

$a = \frac{15}{2}, - \frac{5}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 a - 5 |$
$y = | a + 10 |$
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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