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[tex]|x-7|=5[/tex]ExplanationWe know that we need to write our absolute value equation as [tex]|x-b|=c[/tex]. We also know that the solutions must be [tex]x=2[/tex] and [tex]x=12[/tex]. We need to replace those values for [tex]x[/tex] in our absolute value equation, so we can create a system of equations and find the values of  [tex]b[/tex] and [tex]c[/tex].- For [tex]x=2[/tex][tex]|x-b|=c[/tex][tex]|2-b|=c[/tex] equation (1)- For [tex]x=12[/tex][tex]|x-b|=c[/tex][tex]|12-b|=c[/tex] equation (2)Now we can solve our system of equations step-by-step:Step 1. Replace equation (1) in equation (2)[tex]|12-b|=|2-b|[/tex]Step 2. Square both sides of the equation to get rid of the absolute values[tex]|12-b|=|2-b|[/tex][tex](12-b)^2=(2-b)^2[/tex]Step 3. Use the square of a binomial formula: [tex](a-b)^2=a^2-2ab+b^2[/tex] and solve the equation. For our first binomial, [tex](12-b)^2[/tex], [tex]a=12[/tex] and [tex]b=b[/tex]; for our second binomial, [tex](2-b)^2[/tex], [tex]a=2[/tex] and [tex]b=b[/tex][tex](12-b)^2=(2-b)^2[/tex][tex]12^2-(2)(12)(b)+b^2=2^2-(2)(2)(b)+b^2[/tex][tex]144-24b+b^2=4-4b+b^2[/tex][tex]144-24b=4-4b[/tex][tex]140=20b[/tex][tex]b=\frac{140}{20}[/tex][tex]b=7[/tex] equation (3)Step 4. Replace equation (3) in equation (2) to find the value of [tex]c[/tex][tex]|12-b|=c[/tex][tex]|12-5|=c[/tex][tex]|7|=c[/tex][tex]c=7[/tex]Putting it all together we can conclude that our absolute value equation is [tex]|x-7|=5[/tex]