Question

Which statements are true? Check all that apply.
A. The equation |–x – 4| = 8 will have two solutions.
B. The equation 3.4|0.5x – 42.1| = –20.6 will have one solution.
C. The equation |1/2x-3/4| = 0 will have no solutions.
D. The equation |2x – 10| = –20 will have two solutions.
E. The equation |0.5x – 0.75| + 4.6 = 0.25 will have no solutions.
F. The equation |1/8x-1|= 5 will have infinitely many solutions.

Answer 1

Hello,

A is True

-x-4=8==>x=-12

or -(-x-4)=8==>x=4

B is false

|0.5x-42.1| is always >0

but -20.6/3.4<0 ==> no solution

C: ? not a equation

D is false |2x-10|=-20has no solution

E: ? not an equation

I don't know what you expected!

Answer 2

Answer: (A), (E)

Steps:

(A) The equation has two solutions. Statement True.

$|-x-4|=8\\-x-4=8\implies x=-12\,\,\mbox{, \,or}\,\,-(-x-4)=8\implies x=4$

(B) The equation has no solutions. Statement False.

$3.4\cdot|0.5x-42.1| =-20.6$

No need to solve: the left hand side is always positive while right-hand side is negative, therefore this equation has no solutions in x. Statement False.

(C) it is sufficient to show that the equation has at least one solution to conclude the statement is False:

$|\frac{1}{2}x-\frac{3}{4}| = 0\\\frac{1}{2}x-\frac{3}{4} = 0\\\frac{1}{2}x=\frac{3}{4}\\x = \frac{3}{2}$

(D) Similar argument as with (B): the equation |2x – 10| = –20 has no solutions because its left hand side is always>=0 while its right hand side is negative 20. False.

(E) Statement is True.

It suffices to show:

$|0.5x-0.75|+4.6 = 0.25\\|0.5x-0.75|= 0.25-4.6=-4.35$

from which it is clear that the equation is never satisfied (by any x), similar argument as with (B) and (D). So statement is True.

(F) The equation is two solutions. Statement is False.

$|\frac{1}{8}x-1|= 5\\\frac{1}{8}x-1= 5\\\frac{1}{8}x= 6\\x=48\\\mbox{or}\\-(\frac{1}{8}x-1)= 5\\-\frac{1}{8}x+1= 5\\x= -32$