To solve this problem, we need to compare the slopes of two different lines and determine the constraints on [tex]\( m \)[/tex], based on the condition given in the question.
1. Understand the problem:
- We have two lines in the [tex]\( xy \)[/tex]-plane:
- Line 1: [tex]\( y = mx - 4 \)[/tex]
- Line 2: [tex]\( y = x - 4 \)[/tex]
- We need the slope of Line 1 to be less than the slope of Line 2.
2. Identify the slopes:
- The slope of Line 1 is [tex]\( m \)[/tex].
- The slope of Line 2 is [tex]\( 1 \)[/tex] (since it's in the form [tex]\( y = mx + c \)[/tex] and [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex]).
3. Set up the inequality:
- We need the slope of Line 1 to be less than the slope of Line 2.
- Therefore, [tex]\( m < 1 \)[/tex].
4. Analyze the given choices to determine which values satisfy [tex]\( m < 1 \)[/tex]:
- [tex]\( m = -1 \)[/tex]: This value is less than 1, so it satisfies the inequality [tex]\( m < 1 \)[/tex].
- [tex]\( m = 1 \)[/tex]: This value is not less than 1; it is equal to 1, so it does not satisfy the inequality.
- [tex]\( m < 1 \)[/tex]: This is a direct expression of the inequality we need, so it is correct.
- [tex]\( m > 1 \)[/tex]: This value does not satisfy the inequality since it is greater than 1.
Therefore, the values from the provided choices that must be true about [tex]\( m \)[/tex] are:
[tex]\[ m = -1 \][/tex]
[tex]\[ m < 1 \][/tex]
Additionally, if we need specific examples of [tex]\( m \)[/tex] that satisfy this inequality from a provided list of numerical values, we would find that:
- [tex]\( m = -1 \)[/tex]: satisfies [tex]\( m < 1 \)[/tex]
- [tex]\( m = \frac{999}{1000} \)[/tex]: satisfies [tex]\( m < 1 \)[/tex] (since [tex]\( \frac{999}{1000} \approx 0.999 \)[/tex])
So, the correct values of [tex]\( m \)[/tex] are:
[tex]\[ m = -1 \][/tex]
[tex]\[ \frac{999}{1000} < 1 \text{ (which is approximately 0.999) } \][/tex]
Summing up, the final values satisfying [tex]\( m < 1 \)[/tex] are:
[tex]\[ -1 \][/tex]
[tex]\[ \frac{999}{1000} \][/tex]
