Answer:
1) y = x +17
2) y = (x+5)(x-1)
Step-by-step explanation:
The [absolute value] of a number or expression describes its distance from 0 on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or 0.
For example, −4 and 4 both have an absolute value of 4 because they are each 4 units from 0 on a number line—though they are located in opposite directions from 0 on the number line.
When solving absolute value equations and inequalities, you have to consider both the behavior of absolute value and the properties of equality and inequality.
Because both positive and negative values have a positive absolute value, solving absolute value equations means finding the solution for both the positive and the negative values.
This equation is read “the absolute value of x is equal to five.” The solution is the value(s) that are five units away from 0 on a number line.
You might think of 5 right away; that is one solution to the equation. Notice that −5 is also a solution because −5 is 5 units away from 0 in the opposite direction. So, the solution to this equation is x = −5 or x = 5.
A more complex absolute value problem is solved in a similar fashion. Consider . This equation asks you to find what number plus 5 has an absolute value of 15. Since 15 and −15 both have an absolute value of 15, the absolute value equation is true when the quantity x + 5 is 15 or x + 5 is −15, since |15| = 15 and |−15| = 15. So, you need to find out what value for x will make this expression equal to 15 as well as what value for x will make the expression equal to −15. Solving the two equations you get:
You can check these two solutions in the absolute value equation to see if x = 10 and x = −20 are correct.
A general absolute value equation is written as:
|x - a| = b
Here we will find that the one with only one solution is |x + 17| = 0.
While the one with two solutions is |x + 2| = 3
For the first case, if we know that we have only one solution, then we must have b = 0.
The general equation becomes:
|x - a| = 0
Now we input the desired solution, which is x= -17
|-17 - a| = 0
-17 - a = 0
-17 = a
Then the absolute value equation is: |x + 17| = 0.
Now for the second case, here we have two solutions.
Remember that the general absolute value equation:
|x - a| = b
can be rewritten as:
x - a = b
and
x - a = -b
Now, we input each one of the solutions in each one of these equations. Let's use the positive solution in the equation with the positive b.
1 - a = b
-5 - a = -b
now we can rewrite the second equation to get:
5 + a = b
then we can write:
1 - a = b = 5 + a
1 - a = 5 + a
1 - 5 = 2a
-4 = 2a
-4/2 = a = -2
Now that we know the value of a, we can use any of the two equations to find the value of b.
1 - a = b
1 - (-2) =b
1 + 2 = b
3 = b
The absolute value equation is: |x + 2| = 3
If you want to learn more, you can read:
https://brainly.com/question/1301718
