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Solution - Absolute value equations

Exact form:

a = - \frac{7}{2}, \frac{7}{6}

a=-\frac{7}{2} , \frac{7}{6}

Mixed number form:

a = - 3 \frac{1}{2}, 1 \frac{1}{6}

a=-3\frac{1}{2} , 1\frac{1}{6}

Decimal form:

a = - 3.5, 1.167

a=-3.5 , 1.167

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
$| 2 a - 7 | = | 4 a |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 a - 7 \| = \| 4 a \|$
$x = + y$	$(2 a - 7) = (4 a)$
$x = - y$	$(2 a - 7) = - (4 a)$
$+ x = y$	$(2 a - 7) = (4 a)$
$- x = y$	$- (2 a - 7) = (4 a)$

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:

$\| x \| = \| y \|$	$\| 2 a - 7 \| = \| 4 a \|$
$x = + y$ , $+ x = y$	$(2 a - 7) = (4 a)$
$x = - y$ , $- x = y$	$(2 a - 7) = - (4 a)$

2. Solve the two equations for a

10 additional steps

$(2 a - 7) = 4 a$

Subtract from both sides:

$(2 a - 7) - 4 a = (4 a) - 4 a$

Group like terms:

$(2 a - 4 a) - 7 = (4 a) - 4 a$

Simplify the arithmetic:

$- 2 a - 7 = (4 a) - 4 a$

Simplify the arithmetic:

$- 2 a - 7 = 0$

Add to both sides:

$(- 2 a - 7) + 7 = 0 + 7$

Simplify the arithmetic:

$- 2 a = 0 + 7$

Simplify the arithmetic:

$- 2 a = 7$

Divide both sides by :

$\frac{(- 2 a)}{- 2} = \frac{7}{- 2}$

Cancel out the negatives:

$\frac{2 a}{2} = \frac{7}{- 2}$

Simplify the fraction:

$a = \frac{7}{- 2}$

Move the negative sign from the denominator to the numerator:

$a = \frac{- 7}{2}$

7 additional steps

$(2 a - 7) = - 4 a$

Add to both sides:

$(2 a - 7) + 7 = (- 4 a) + 7$

Simplify the arithmetic:

$2 a = (- 4 a) + 7$

Add to both sides:

$(2 a) + 4 a = ((- 4 a) + 7) + 4 a$

Simplify the arithmetic:

$6 a = ((- 4 a) + 7) + 4 a$

Group like terms:

$6 a = (- 4 a + 4 a) + 7$

Simplify the arithmetic:

$6 a = 7$

Divide both sides by :

$\frac{(6 a)}{6} = \frac{7}{6}$

Simplify the fraction:

$a = \frac{7}{6}$

3. List the solutions

$a = - \frac{7}{2}, \frac{7}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 a - 7 |$
$y = | 4 a |$
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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