Question

How many solutions does the equation a|x+b|+c=d have if a>0 and c=d?then if a <0and c>d?

Answer 1

1.

`a|x+b|+c=d \ \ \ |-c \\ a|x+b|=d-c \\ \\ c=d \hbox{ so } d-c=0 \\ \\ a|x+b|=0 \ \ \ |/ a, \ a>0 \\ |x+b|=(0)/(a) \\ |x+b|=0 \\ \\ \hbox{if } |x|=0, \hbox{ then } x=0 \\ \\ x+b=0 \\ x=-b`

If a>0 and c=d, the equation has one solution.

2.

`a|x+b|+c=d \ \ \ |-c \\ a|x+b|=d-c \ \ \ |/ a, a <0 \\ |x+b|=(d-c)/(a) \\ \\ c>d \hbox{ so } d-c<0 \\ a<0 \\ \hbox{a negative number divided by a negative number is positive number so} \\ (d-c)/(a) > 0 \\ \hbox{if } |x|=a, \ a>0, \hbox{ then } x=a \ \lor \ x=-a \\ \\ |x+b|=(d-c)/(a) \\ x+b=(d-c)/(a) \ \lor \ x+b=-(d-c)/(a) \\ x=(d-c)/(a)-b \ \lor \ x=-(d-c)/(a)-b`

If a<0 and c>d, the equation has two solutions.