To determine which option must be true about [tex]\( m \)[/tex] 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], let's analyze the slopes of both lines.
1. Identifying the slopes:
- The equation [tex]\( y = mx - 4 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of this line is [tex]\( m \)[/tex].
- The equation [tex]\( y = x - 4 \)[/tex] is also in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope is the coefficient of [tex]\( x \)[/tex]. Here, the slope is [tex]\( 1 \)[/tex].
2. Given Condition:
- According to the problem, the slope of the line [tex]\( y = mx - 4 \)[/tex] is less than the slope of the line [tex]\( y = x - 4 \)[/tex]. This translates to the inequality:
[tex]\[
m < 1
\][/tex]
3. Interpretation of the inequality:
- We need to select the option that correctly represents this inequality. The inequality [tex]\( m < 1 \)[/tex] indicates that [tex]\( m \)[/tex] must be less than [tex]\( 1 \)[/tex].
Given this, let's review the options:
- A. [tex]\( m = -1 \)[/tex]: While this is a specific value of [tex]\( m \)[/tex], it does not necessarily cover all values where [tex]\( m < 1 \)[/tex].
- B. [tex]\( m = 1 \)[/tex]: This is the equality case and does not satisfy [tex]\( m < 1 \)[/tex].
- C. [tex]\( m < 1 \)[/tex]: This directly represents the condition given by the problem.
- D. [tex]\( m > 1 \)[/tex]: This contradicts the inequality [tex]\( m < 1 \)[/tex].
Thus, the correct answer that must be true about [tex]\( m \)[/tex] is:
[tex]\[
\boxed{m < 1}
\][/tex]
