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Solution - Absolute value equations

Exact form:

a = - 13, 4

a=-13 , 4

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
$| 5 a - 3 | = | 3 a - 29 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 a - 3 \| = \| 3 a - 29 \|$
$x = + y$	$(5 a - 3) = (3 a - 29)$
$x = - y$	$(5 a - 3) = - (3 a - 29)$
$+ x = y$	$(5 a - 3) = (3 a - 29)$
$- x = y$	$- (5 a - 3) = (3 a - 29)$

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:

$\| x \| = \| y \|$	$\| 5 a - 3 \| = \| 3 a - 29 \|$
$x = + y$ , $+ x = y$	$(5 a - 3) = (3 a - 29)$
$x = - y$ , $- x = y$	$(5 a - 3) = - (3 a - 29)$

2. Solve the two equations for a

11 additional steps

$(5 a - 3) = (3 a - 29)$

Subtract from both sides:

$(5 a - 3) - 3 a = (3 a - 29) - 3 a$

Group like terms:

$(5 a - 3 a) - 3 = (3 a - 29) - 3 a$

Simplify the arithmetic:

$2 a - 3 = (3 a - 29) - 3 a$

Group like terms:

$2 a - 3 = (3 a - 3 a) - 29$

Simplify the arithmetic:

$2 a - 3 = - 29$

Add to both sides:

$(2 a - 3) + 3 = - 29 + 3$

Simplify the arithmetic:

$2 a = - 29 + 3$

Simplify the arithmetic:

$2 a = - 26$

Divide both sides by :

$\frac{(2 a)}{2} = \frac{- 26}{2}$

Simplify the fraction:

$a = \frac{- 26}{2}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(- 13 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = - 13$

12 additional steps

$(5 a - 3) = - (3 a - 29)$

Expand the parentheses:

$(5 a - 3) = - 3 a + 29$

Add to both sides:

$(5 a - 3) + 3 a = (- 3 a + 29) + 3 a$

Group like terms:

$(5 a + 3 a) - 3 = (- 3 a + 29) + 3 a$

Simplify the arithmetic:

$8 a - 3 = (- 3 a + 29) + 3 a$

Group like terms:

$8 a - 3 = (- 3 a + 3 a) + 29$

Simplify the arithmetic:

$8 a - 3 = 29$

Add to both sides:

$(8 a - 3) + 3 = 29 + 3$

Simplify the arithmetic:

$8 a = 29 + 3$

Simplify the arithmetic:

$8 a = 32$

Divide both sides by :

$\frac{(8 a)}{8} = \frac{32}{8}$

Simplify the fraction:

$a = \frac{32}{8}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(4 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$a = 4$

3. List the solutions

$a = - 13, 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 a - 3 |$
$y = | 3 a - 29 |$
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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