Write an absolute value equation that has the solutions x=8 and x=18.

|x - 13| = 5

Step-by-step explanation:

Think of absolute value as a distance. For example, |5| can be written as |5 - 0| = 5 since the distance between 5 and zero on the number line is 5. Also, |-5| which is the same as |-5 - 0| is also 5 since the distance between -5 and zero on the number line is 5.

Since you want an absolute value equation with solutions 8 and 18, think of which number is the same distance from both 8 and 18 on the number line? Find the average of 8 and 18, (8 + 18)/2 = 13. 13 is the same distance from both 8 and 18. 13 is 5 units away from both 8 and 18. Now you need an equation that is looking for the numbers x which are 5 units distance from 13.

|x - 13| = 5

Here we want to find an absolute value equation only by knowing its solutions. We will find that the function is |x - 13| = 5

We want an absolute value equation with solutions x = 8 and x = 18.

Remember that a general absolute value equation can be written as:

|x - a| = b

We can rewrite this in two equations:

x - a = b

x - a = -b

Now let's replace the known solutions in the image, we should use the larger value of x in the equation with the positive b.

18 - a = b

8 - a = -b

This is a system of equations, to solve this, we can rewrite the second equation as:

8 -a = -b

-8 + a = b

Now we can replace this in the first equation:

18 - a = b = -8 + a

18 - a = -8 + a

18 + 8  = a + a

26 = 2a

26/2 = a = 13

Now we know the value of a, then we can use the equation:

18 - 13 = b = 5

Then the absolute value equation is:

|x - 13| = 5

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