Q:

Help Which of the following graphs represents the solution of the absolute value inequality |y + 8| < 3?Please select the best answer from the choices provided

Accepted Solution

A:
Answer:-11 < y <-5 which is option BStep-by-step explanation:When dealing with absolute value expressions, one needs to always analyze separately each possible situation: 1) When the expression inside the absolute value symbols is larger or equal zero, and 2) when the expression inside the absolute value symbols is less than zero, so we can get rid of the absolute symbol and be free to move variables in order to solve the inequality.The expression inside the absolute value bars in this case is: y + 8, so we study what happens when1) y + 8 is larger of equal zeroIn this case the absolute value of y + 8 remains without change as we remove the absolute bar symbol, so we get:[tex]|y+8|=y+8\\[/tex], then solving for the inequality (isolating the variable y on one side of the inequality symbol):[tex]y+8<3\\y<3-8\\y<-5[/tex]which tells us that we need to consider the real values that are smaller than -5, and that is those to the left of "-5" on the number line.2) y = 8 is less than zeroIn this case, the absolute value of y + 8 becomes the opposite of it, that is: [tex]|y+8|=-(y+8)=-y-8[/tex]Therefore, replacing this in the original inequality and solving for the variable y gives us:[tex]|y+8|<3\\-y-8<3\\-8<3+y\\-8-3<y\\-11<y[/tex]which tess us that we need to consider the real values larger than -11 which are those to the right of -11 on the number line.The combination of both results (from parts 1 and 2) gives us those real numbers y that are larger than -11 (to the right of -11), and at the same time smaller than -5 (to the left of -5). This is represented by the graph of option "B".