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Solution - Absolute value equations

Exact form:

a = 16, - \frac{2}{5}

a=16 , -\frac{2}{5}

Decimal form:

a = 16, - 0.4

a=16 , -0.4

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
$| 2 a + 9 | = | 3 a - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 a + 9 \| = \| 3 a - 7 \|$
$x = + y$	$(2 a + 9) = (3 a - 7)$
$x = - y$	$(2 a + 9) = - (3 a - 7)$
$+ x = y$	$(2 a + 9) = (3 a - 7)$
$- x = y$	$- (2 a + 9) = (3 a - 7)$

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:

$\| x \| = \| y \|$	$\| 2 a + 9 \| = \| 3 a - 7 \|$
$x = + y$ , $+ x = y$	$(2 a + 9) = (3 a - 7)$
$x = - y$ , $- x = y$	$(2 a + 9) = - (3 a - 7)$

2. Solve the two equations for a

10 additional steps

$(2 a + 9) = (3 a - 7)$

Subtract from both sides:

$(2 a + 9) - 3 a = (3 a - 7) - 3 a$

Group like terms:

$(2 a - 3 a) + 9 = (3 a - 7) - 3 a$

Simplify the arithmetic:

$- a + 9 = (3 a - 7) - 3 a$

Group like terms:

$- a + 9 = (3 a - 3 a) - 7$

Simplify the arithmetic:

$- a + 9 = - 7$

Subtract from both sides:

$(- a + 9) - 9 = - 7 - 9$

Simplify the arithmetic:

$- a = - 7 - 9$

Simplify the arithmetic:

$- a = - 16$

Multiply both sides by :

$- a \cdot - 1 = - 16 \cdot - 1$

Remove the one(s):

$a = - 16 \cdot - 1$

Simplify the arithmetic:

$a = 16$

10 additional steps

$(2 a + 9) = - (3 a - 7)$

Expand the parentheses:

$(2 a + 9) = - 3 a + 7$

Add to both sides:

$(2 a + 9) + 3 a = (- 3 a + 7) + 3 a$

Group like terms:

$(2 a + 3 a) + 9 = (- 3 a + 7) + 3 a$

Simplify the arithmetic:

$5 a + 9 = (- 3 a + 7) + 3 a$

Group like terms:

$5 a + 9 = (- 3 a + 3 a) + 7$

Simplify the arithmetic:

$5 a + 9 = 7$

Subtract from both sides:

$(5 a + 9) - 9 = 7 - 9$

Simplify the arithmetic:

$5 a = 7 - 9$

Simplify the arithmetic:

$5 a = - 2$

Divide both sides by :

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

Simplify the fraction:

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

3. List the solutions

$a = 16, - \frac{2}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 a + 9 |$
$y = | 3 a - 7 |$
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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