Answer :
To determine which condition must be true about [tex]\( m \)[/tex], we first need to understand the concept of slope in the context of linear equations in the [tex]\( xy \)[/tex]-plane.
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
Given the two lines:
1. [tex]\( y = mx - 4 \)[/tex]
2. [tex]\( y = x - 4 \)[/tex]
We can identify the slopes of these lines:
- For the first line, [tex]\( y = mx - 4 \)[/tex], the slope is [tex]\( m \)[/tex].
- For the second line, [tex]\( y = x - 4 \)[/tex], the slope is [tex]\( 1 \)[/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]. Mathematically, this can be written as:
[tex]\[ m < 1 \][/tex]
This inequality indicates that the value of [tex]\( m \)[/tex] must be less than 1.
Now, let's review the given options to see which one fits this condition:
- [tex]\( m = -1 \)[/tex]: This is less than 1, so this condition could be true.
- [tex]\( m = 1 \)[/tex]: This is equal to 1, not less than 1, so this condition is not true.
- [tex]\( m < 1 \)[/tex]: This directly matches our derived condition, so it must be true.
- [tex]\( m > 1 \)[/tex]: This is greater than 1, so this condition is not true.
Therefore, the option that must be true is:
[tex]\[ m < 1 \][/tex]
The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
Given the two lines:
1. [tex]\( y = mx - 4 \)[/tex]
2. [tex]\( y = x - 4 \)[/tex]
We can identify the slopes of these lines:
- For the first line, [tex]\( y = mx - 4 \)[/tex], the slope is [tex]\( m \)[/tex].
- For the second line, [tex]\( y = x - 4 \)[/tex], the slope is [tex]\( 1 \)[/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]. Mathematically, this can be written as:
[tex]\[ m < 1 \][/tex]
This inequality indicates that the value of [tex]\( m \)[/tex] must be less than 1.
Now, let's review the given options to see which one fits this condition:
- [tex]\( m = -1 \)[/tex]: This is less than 1, so this condition could be true.
- [tex]\( m = 1 \)[/tex]: This is equal to 1, not less than 1, so this condition is not true.
- [tex]\( m < 1 \)[/tex]: This directly matches our derived condition, so it must be true.
- [tex]\( m > 1 \)[/tex]: This is greater than 1, so this condition is not true.
Therefore, the option that must be true is:
[tex]\[ m < 1 \][/tex]
