To determine which condition must be true about [tex]\( m \)[/tex], we need to compare the slopes of the two given lines in the [tex]\( xy \)[/tex]-plane:
1. The first line is given by the equation:
[tex]\[
y = mx - 4
\][/tex]
The slope of this line is [tex]\( m \)[/tex].
2. The second line is given by the equation:
[tex]\[
y = x - 4
\][/tex]
The slope of this line is [tex]\( 1 \)[/tex].
We are told that the slope of the first line must be less than the slope of the second line. Therefore, we set up the inequality:
[tex]\[
m < 1
\][/tex]
This inequality tells us that the value of [tex]\( m \)[/tex] must be less than [tex]\( 1 \)[/tex].
Evaluating the given options:
- [tex]\( m = -1 \)[/tex]: [tex]\( m < 1 \)[/tex] (This satisfies the condition)
- [tex]\( m = 1 \)[/tex]: [tex]\( m < 1 \)[/tex] (This does not satisfy the condition)
- [tex]\( m < 1 \)[/tex]: This is the inequality we derived and is the condition that must be true.
- [tex]\( m > 1 \)[/tex]: [tex]\( m < 1 \)[/tex] (This does not satisfy the condition)
Hence, the statement that must be true about [tex]\( m \)[/tex] is:
[tex]\[
m < 1
\][/tex]
