Question

Write the absolute value equation if it has the following solutions. Hint: Your equation should be written as |x−b| =c. (Here b and c are some numbers.) Chapter Reference b Two solutions: x=2, x=12.

Answer 1

Answer

$|x-7|=5$

Explanation

We know that we need to write our absolute value equation as $|x-b|=c$. We also know that the solutions must be $x=2$ and $x=12$. We need to replace those values for $x$ in our absolute value equation, so we can create a system of equations and find the values of  $b$ and $c$.

- For $x=2$

$|x-b|=c$

$|2-b|=c$ equation (1)

- For $x=12$

$|x-b|=c$

$|12-b|=c$ equation (2)

Now we can solve our system of equations step-by-step:

Step 1. Replace equation (1) in equation (2)

$|12-b|=|2-b|$

Step 2. Square both sides of the equation to get rid of the absolute values

$|12-b|=|2-b|$

$(12-b)^2=(2-b)^2$

Step 3. Use the square of a binomial formula: $(a-b)^2=a^2-2ab+b^2$ and solve the equation.

For our first binomial, $(12-b)^2$, $a=12$ and $b=b$; for our second binomial, $(2-b)^2$, $a=2$ and $b=b$

$(12-b)^2=(2-b)^2$

$12^2-(2)(12)(b)+b^2=2^2-(2)(2)(b)+b^2$

$144-24b+b^2=4-4b+b^2$

$144-24b=4-4b$

$140=20b$

$b=\frac{140}{20}$

$b=7$ equation (3)

Step 4. Replace equation (3) in equation (2) to find the value of $c$

$|12-b|=c$

$|12-5|=c$

$|7|=c$

$c=7$

Putting it all together we can conclude that our absolute value equation is $|x-7|=5$