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Solution - Absolute value equations

Exact form:

x = 3, \frac{15}{11}

x=3 , \frac{15}{11}

Mixed number form:

x = 3, 1 \frac{4}{11}

x=3 , 1\frac{4}{11}

Decimal form:

x = 3, 1.364

x=3 , 1.364

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
$| 7 x - 12 | = | 4 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x - 12 \| = \| 4 x - 3 \|$
$x = + y$	$(7 x - 12) = (4 x - 3)$
$x = - y$	$(7 x - 12) = - (4 x - 3)$
$+ x = y$	$(7 x - 12) = (4 x - 3)$
$- x = y$	$- (7 x - 12) = (4 x - 3)$

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:

$\| x \| = \| y \|$	$\| 7 x - 12 \| = \| 4 x - 3 \|$
$x = + y$ , $+ x = y$	$(7 x - 12) = (4 x - 3)$
$x = - y$ , $- x = y$	$(7 x - 12) = - (4 x - 3)$

2. Solve the two equations for x

11 additional steps

$(7 x - 12) = (4 x - 3)$

Subtract from both sides:

$(7 x - 12) - 4 x = (4 x - 3) - 4 x$

Group like terms:

$(7 x - 4 x) - 12 = (4 x - 3) - 4 x$

Simplify the arithmetic:

$3 x - 12 = (4 x - 3) - 4 x$

Group like terms:

$3 x - 12 = (4 x - 4 x) - 3$

Simplify the arithmetic:

$3 x - 12 = - 3$

Add to both sides:

$(3 x - 12) + 12 = - 3 + 12$

Simplify the arithmetic:

$3 x = - 3 + 12$

Simplify the arithmetic:

$3 x = 9$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{9}{3}$

Simplify the fraction:

$x = \frac{9}{3}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = 3$

10 additional steps

$(7 x - 12) = - (4 x - 3)$

Expand the parentheses:

$(7 x - 12) = - 4 x + 3$

Add to both sides:

$(7 x - 12) + 4 x = (- 4 x + 3) + 4 x$

Group like terms:

$(7 x + 4 x) - 12 = (- 4 x + 3) + 4 x$

Simplify the arithmetic:

$11 x - 12 = (- 4 x + 3) + 4 x$

Group like terms:

$11 x - 12 = (- 4 x + 4 x) + 3$

Simplify the arithmetic:

$11 x - 12 = 3$

Add to both sides:

$(11 x - 12) + 12 = 3 + 12$

Simplify the arithmetic:

$11 x = 3 + 12$

Simplify the arithmetic:

$11 x = 15$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{15}{11}$

Simplify the fraction:

$x = \frac{15}{11}$

3. List the solutions

$x = 3, \frac{15}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 x - 12 |$
$y = | 4 x - 3 |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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