Answered

In the [tex]$xy$[/tex]-plane, the slope of the line [tex]$y=mx-4$[/tex] is less than the slope of the line [tex][tex]$y=x-4$[/tex][/tex]. Which of the following must be true about [tex]$m$[/tex]?

A. [tex]$m=-1$[/tex]
B. [tex][tex]$m=1$[/tex][/tex]
C. [tex]$m\ \textless \ 1$[/tex]
D. [tex]$m\ \textgreater \ 1$[/tex]



Answer :

To determine the correct answer for the given question, we need to compare the slopes of the two lines provided in the equations [tex]\( y = mx - 4 \)[/tex] and [tex]\( y = x - 4 \)[/tex].

1. Identify the slopes of each line:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line.
- For the line [tex]\( y = mx - 4 \)[/tex], the slope is [tex]\( m \)[/tex].
- For the line [tex]\( y = x - 4 \)[/tex], the slope is 1.

2. Comparing the slopes:
- The problem states that the slope of the line [tex]\( y = mx - 4 \)[/tex] must be less than the slope of the line [tex]\( y = x - 4 \)[/tex].
- Therefore, we need [tex]\( m \)[/tex] to be less than 1.

Mathematically, this inequality can be expressed as:
[tex]\[ m < 1 \][/tex]

3. Analyze each option:
- [tex]\( m = -1 \)[/tex]: This value indeed satisfies [tex]\( m < 1 \)[/tex].
- [tex]\( m = 1 \)[/tex]: This value does NOT satisfy [tex]\( m < 1 \)[/tex] because [tex]\( 1 \)[/tex] is not less than [tex]\( 1 \)[/tex].
- [tex]\( m < 1 \)[/tex]: This is directly the inequality that we need.
- [tex]\( m > 1 \)[/tex]: This value does NOT satisfy [tex]\( m < 1 \)[/tex].

Given our analysis:

The correct option is:
[tex]\[ \boxed{m < 1} \][/tex]