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In the [tex]\(xy\)[/tex]-plane, the slope of the line [tex]\(y = mx - 4\)[/tex] is less than the slope of the line [tex]\(y = x - 4\)[/tex]. Which of the following must be true about [tex]\(m\)[/tex]?

A. [tex]\(m \ \textless \ -1\)[/tex]
B. [tex]\(m \ \textless \ 1\)[/tex]
C. [tex]\(m = 1\)[/tex]
D. [tex]\(m \ \textgreater \ 1\)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to analyze the slopes of the given lines in the [tex]\(xy\)[/tex]-plane and compare them. We are given two lines:

1. [tex]\( y = mx - 4 \)[/tex]
2. [tex]\( y = x - 4 \)[/tex]

### Step-by-Step Solution:

1. Identify the slopes:
- The slope of a line in the form [tex]\( y = mx + b \)[/tex] is given by the coefficient of [tex]\( x \)[/tex], which is [tex]\( m \)[/tex].
- The slope of the line [tex]\( y = mx - 4 \)[/tex] is [tex]\( m \)[/tex].
- The slope of the line [tex]\( y = x - 4 \)[/tex] is [tex]\( 1 \)[/tex] since it can be written as [tex]\( y = 1x - 4 \)[/tex].

2. Compare the slopes:
- 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].

Therefore, we can write the inequality:
[tex]\[ m < 1 \][/tex]

3. Interpret the inequality:
- This inequality [tex]\( m < 1 \)[/tex] means that the value of [tex]\( m \)[/tex] must be less than [tex]\( 1 \)[/tex].

### Conclusion:
The correct interpretation of the question is that for the slope of the line [tex]\( y = mx - 4 \)[/tex] to be less than the slope of the line [tex]\( y = x - 4 \)[/tex], the value of [tex]\( m \)[/tex] must be less than [tex]\( 1 \)[/tex].

Thus, the answer is:
[tex]\[ m < 1 \][/tex]

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