To determine which statement about [tex]\( m \)[/tex] is true, we need to analyze the slopes of the lines given in the equations [tex]\( y = mx - 4 \)[/tex] and [tex]\( y = x - 4 \)[/tex].
First, recall that the slope of a line [tex]\( y = mx + b \)[/tex] is given by the coefficient [tex]\( m \)[/tex] of [tex]\( x \)[/tex].
1. For the line [tex]\( y = x - 4 \)[/tex]:
- The slope [tex]\( m_1 \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is 1. Thus, [tex]\( m_1 = 1 \)[/tex].
2. For the line [tex]\( y = mx - 4 \)[/tex]:
- The slope [tex]\( m_2 \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is [tex]\( m \)[/tex].
The problem states that the slope of the line [tex]\( y = mx - 4 \)[/tex] is less than the slope of the line [tex]\( y = x - 4 \)[/tex]. Therefore, we have:
[tex]\[ m < 1 \][/tex]
To restate, we need [tex]\( m \)[/tex] to be less than 1 for the slope of the line [tex]\( y = mx - 4 \)[/tex] to be smaller than the slope of the line [tex]\( y = x - 4 \)[/tex].
Now, let's examine the given choices:
1. [tex]\( m = -1 \)[/tex]
- [tex]\( -1 \)[/tex] is less than 1, so this statement is true.
2. [tex]\( m = 1 \)[/tex]
- [tex]\( 1 \)[/tex] is not less than 1, so this statement is false.
3. [tex]\( m < 1 \)[/tex]
- This is directly what we deduced from the inequality [tex]\( m < 1 \)[/tex], so this statement is true.
4. [tex]\( m > 1 \)[/tex]
- [tex]\( m \)[/tex] being greater than 1 contradicts the inequality [tex]\( m < 1 \)[/tex], so this statement is false.
Therefore, the correct choice is:
[tex]\[ \boxed{m < 1} \][/tex]
