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Solution - Absolute value equations

Exact form:

x = - \frac{29}{2}, \frac{3}{4}

x=-\frac{29}{2} , \frac{3}{4}

Mixed number form:

x = - 14 \frac{1}{2}, \frac{3}{4}

x=-14\frac{1}{2} , \frac{3}{4}

Decimal form:

x = - 14.5, 0.75

x=-14.5 , 0.75

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
$| 5 x - 19 | = | 7 x + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 19 \| = \| 7 x + 10 \|$
$x = + y$	$(5 x - 19) = (7 x + 10)$
$x = - y$	$(5 x - 19) = - (7 x + 10)$
$+ x = y$	$(5 x - 19) = (7 x + 10)$
$- x = y$	$- (5 x - 19) = (7 x + 10)$

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:

$\| x \| = \| y \|$	$\| 5 x - 19 \| = \| 7 x + 10 \|$
$x = + y$ , $+ x = y$	$(5 x - 19) = (7 x + 10)$
$x = - y$ , $- x = y$	$(5 x - 19) = - (7 x + 10)$

2. Solve the two equations for x

11 additional steps

$(5 x - 19) = (7 x + 10)$

Subtract from both sides:

$(5 x - 19) - 7 x = (7 x + 10) - 7 x$

Group like terms:

$(5 x - 7 x) - 19 = (7 x + 10) - 7 x$

Simplify the arithmetic:

$- 2 x - 19 = (7 x + 10) - 7 x$

Group like terms:

$- 2 x - 19 = (7 x - 7 x) + 10$

Simplify the arithmetic:

$- 2 x - 19 = 10$

Add to both sides:

$(- 2 x - 19) + 19 = 10 + 19$

Simplify the arithmetic:

$- 2 x = 10 + 19$

Simplify the arithmetic:

$- 2 x = 29$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{29}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{29}{- 2}$

Simplify the fraction:

$x = \frac{29}{- 2}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 29}{2}$

12 additional steps

$(5 x - 19) = - (7 x + 10)$

Expand the parentheses:

$(5 x - 19) = - 7 x - 10$

Add to both sides:

$(5 x - 19) + 7 x = (- 7 x - 10) + 7 x$

Group like terms:

$(5 x + 7 x) - 19 = (- 7 x - 10) + 7 x$

Simplify the arithmetic:

$12 x - 19 = (- 7 x - 10) + 7 x$

Group like terms:

$12 x - 19 = (- 7 x + 7 x) - 10$

Simplify the arithmetic:

$12 x - 19 = - 10$

Add to both sides:

$(12 x - 19) + 19 = - 10 + 19$

Simplify the arithmetic:

$12 x = - 10 + 19$

Simplify the arithmetic:

$12 x = 9$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{9}{12}$

Simplify the fraction:

$x = \frac{9}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 3)}{(4 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{4}$

3. List the solutions

$x = - \frac{29}{2}, \frac{3}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 19 |$
$y = | 7 x + 10 |$
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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