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Solution - Absolute value equations

Exact form:

a = 14, \frac{4}{7}

a=14 , \frac{4}{7}

Decimal form:

a = 14, 0.571

a=14 , 0.571

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
$| 3 a + 5 | = | 4 a - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 a + 5 \| = \| 4 a - 9 \|$
$x = + y$	$(3 a + 5) = (4 a - 9)$
$x = - y$	$(3 a + 5) = - (4 a - 9)$
$+ x = y$	$(3 a + 5) = (4 a - 9)$
$- x = y$	$- (3 a + 5) = (4 a - 9)$

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:

$\| x \| = \| y \|$	$\| 3 a + 5 \| = \| 4 a - 9 \|$
$x = + y$ , $+ x = y$	$(3 a + 5) = (4 a - 9)$
$x = - y$ , $- x = y$	$(3 a + 5) = - (4 a - 9)$

2. Solve the two equations for a

10 additional steps

$(3 a + 5) = (4 a - 9)$

Subtract from both sides:

$(3 a + 5) - 4 a = (4 a - 9) - 4 a$

Group like terms:

$(3 a - 4 a) + 5 = (4 a - 9) - 4 a$

Simplify the arithmetic:

$- a + 5 = (4 a - 9) - 4 a$

Group like terms:

$- a + 5 = (4 a - 4 a) - 9$

Simplify the arithmetic:

$- a + 5 = - 9$

Subtract from both sides:

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

Simplify the arithmetic:

$- a = - 9 - 5$

Simplify the arithmetic:

$- a = - 14$

Multiply both sides by :

$- a \cdot - 1 = - 14 \cdot - 1$

Remove the one(s):

$a = - 14 \cdot - 1$

Simplify the arithmetic:

$a = 14$

10 additional steps

$(3 a + 5) = - (4 a - 9)$

Expand the parentheses:

$(3 a + 5) = - 4 a + 9$

Add to both sides:

$(3 a + 5) + 4 a = (- 4 a + 9) + 4 a$

Group like terms:

$(3 a + 4 a) + 5 = (- 4 a + 9) + 4 a$

Simplify the arithmetic:

$7 a + 5 = (- 4 a + 9) + 4 a$

Group like terms:

$7 a + 5 = (- 4 a + 4 a) + 9$

Simplify the arithmetic:

$7 a + 5 = 9$

Subtract from both sides:

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

Simplify the arithmetic:

$7 a = 9 - 5$

Simplify the arithmetic:

$7 a = 4$

Divide both sides by :

$\frac{(7 a)}{7} = \frac{4}{7}$

Simplify the fraction:

$a = \frac{4}{7}$

3. List the solutions

$a = 14, \frac{4}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 a + 5 |$
$y = | 4 a - 9 |$
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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