Answer:
Question 1: y = x + 4 if x>5
Question 2: y = x
Question 3: y = -x-4
Question 4: y = 3x-6
Step-by-step explanation:
Ok, I will admit, these questions are a bit confusing, but I will try my best.
Question 1: y = |x-3| + |x+2| - |x-5| if x>5. So, let's plug in a number greater than 5 - 6. y = |6-3| + |6+2| - |6-5|. Inside the absolute value, every number is positive, which pretty much renders the absolute value symbol useless. So, you can take the equation and change the absolute value lines into parentheses. y = (x-3) + (x+2) - (x-5). In order to get rid of the parentheses, you have to distribute any subtraction symbols which is in front of x-5. This makes our equation y = x - 3 + x + 2 -x + 5. Combine like terms to get y = x + 4 if x>5.
Question 2: y = |x-3| + |x+2| - |x-5| if -2<x<3. Let's plug in a number between -2 and 3 so 0. y = |0-3| + |0+2| - |0-5|. x-3 becomes a negative number, x+2 becomes a positive number, and x-5 becomes a negative number. Just like question 1, we can leave the x+2 alone since it's a positive number, but we have to change the x-3 and the x-5. So we'll take the equation and change the absolute value to parentheses, but for the values that are negative, we have to put a negative sign in front of to make it positive. Our equation looks like y = -(x-3) + (x+2) - -(x-5). Since there are two negatives in front of x-5, we can just make it a plus sign. We distribute the negative to x-3 to make our equation y = -x+3 + x+2 + x-5. Combine like terms to get y = x
Question 3: y = |x-3| + |x+2| - |x-5| if x<-2. Let's plug in -3 since it's less than -2. y = |-3-3| + |-3+2| - |-3-5|. All three parts of the function are negative numbers, so when we change it to parentheses, all parts will have a negative to distribute. Our equation looks like y = -(x-3) -(x+2) - -(x-5). Since there are two negatives in front of x-5, we can just make it a plus sign. We distribute the negative to x-3 and x+2 to make our equation y = -x+3 + -x-2 + x-5. Combine like terms to get y = -x-4.
Question 4: y = |x-3| + |x+2| - |x-5| if 3<x<5. Let's plug in 4 since it's in between 3 and 5. y = |4-3| + |4+2| - |4-5|. Only x-5 is negative so that's the only one we need to put a negative in front of when we change it to parentheses. Our equation looks like y = (x-3) + (x+2) - -(x-5). Since there are two negatives in front of x-5, we can just make it a plus sign. Our equation is now y = x-3 + x+2 + x-5. Combine like terms to get y = 3x-6.
Hope this helps! :)
