To determine which condition must be true about [tex]\( m \)[/tex], we need to compare the slopes of the two lines given in the question.
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line.
1. Slope of the line [tex]\( y = mx - 4 \)[/tex]:
- In this equation, [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex].
- Therefore, the slope of this line is [tex]\( m \)[/tex].
2. Slope of the line [tex]\( y = x - 4 \)[/tex]:
- For this equation, notice that the coefficient of [tex]\( x \)[/tex] is 1.
- Thus, the slope of this line is 1.
We are given that the slope of the line [tex]\( y = mx - 4 \)[/tex] is less than the slope of the line [tex]\( y = x - 4 \)[/tex]. Mathematically, this can be written as:
[tex]\[ m < 1 \][/tex]
Now, let's analyze each of the conditions provided:
- [tex]\( m = -1 \)[/tex]: Here, [tex]\(-1 < 1\)[/tex] is true.
- [tex]\( m = 1 \)[/tex]: Here, [tex]\(1 < 1\)[/tex] is false.
- [tex]\( m < 1 \)[/tex]: This is the exact inequality we need.
- [tex]\( m > 1 \)[/tex]: Here, [tex]\(m > 1\)[/tex] would imply that [tex]\( m \)[/tex] is not less than 1, so this is false.
Among the given options, the correct condition that must be true about [tex]\( m \)[/tex] is:
[tex]\[ m < 1 \][/tex]
Thus, the correct answer is [tex]\( m < 1 \)[/tex].
