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Solution - Absolute value equations

Exact form:

x = 7

x=7

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| x + 5 | + | x - 19 | = 0$

Add $- | x - 19 |$ to both sides of the equation:

$| x + 5 | + | x - 19 | - | x - 19 | = - | x - 19 |$

Simplify the arithmetic

$| x + 5 | = - | x - 19 |$

2. 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
$| x + 5 | = - | x - 19 |$
without the absolute value bars:

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

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:

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

3. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$2 x + 5 = (- x + 19) + x$

Group like terms:

$2 x + 5 = (- x + x) + 19$

Simplify the arithmetic:

$2 x + 5 = 19$

Subtract from both sides:

$(2 x + 5) - 5 = 19 - 5$

Simplify the arithmetic:

$2 x = 19 - 5$

Simplify the arithmetic:

$2 x = 14$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{14}{2}$

Simplify the fraction:

$x = \frac{14}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 7$

6 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(x + 5) = x - 19$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$5 = (x - 19) - x$

Group like terms:

$5 = (x - x) - 19$

Simplify the arithmetic:

$5 = - 19$

The statement is false:

$5 = - 19$

The equation is false so it has no solution.

4. List the solutions

$x = 7$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x + 5 |$
$y = - | x - 19 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved