Answered

Discover how IDNLearn.com can help you find the answers you need quickly and easily. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.

Which statements are true about the linear inequality [tex]\( y \ \textgreater \ \frac{3}{4}x - 2 \)[/tex]? Select three options.

A. The slope of the line is [tex]\(-2\)[/tex].
B. The graph of [tex]\( y \ \textgreater \ \frac{3}{4}x - 2 \)[/tex] is a dashed line.
C. The area below the line is shaded.
D. One solution to the inequality is [tex]\((0,0)\)[/tex].
E. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\((0,-2)\)[/tex].

Sagot :

GinnyAnsweridn
Let's analyze the given linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] step-by-step to determine which statements are true.

1. The slope of the line is -2:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In the given inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
- Thus, the statement saying the slope of the line is -2 is false.

2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line:
- For inequalities of the form [tex]\( y > mx + b \)[/tex] or [tex]\( y < mx + b \)[/tex], the boundary line is dashed because the inequality does not include the equality (i.e., it is a strict inequality).
- Therefore, this statement is true.

3. The area below the line is shaded:
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] indicates that the region above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is the solution region.
- So, the area below the line is not shaded.
- This statement is false.

4. One solution to the inequality is [tex]\( (0,0) \)[/tex]:
- Plugging [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 0 > \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ 0 > -2 \][/tex]
- Since this inequality holds true, [tex]\( (0,0) \)[/tex] is indeed a solution.
- Thus, this statement is true.

5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]:
- The y-intercept is the point where [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y = \frac{3}{4} x - 2 \)[/tex]:
[tex]\[ y = \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ y = -2 \][/tex]
- Therefore, the y-intercept is at [tex]\( (0, -2) \)[/tex].
- This statement is true as well.

So, the three true statements are:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- One solution to the inequality is [tex]\( (0,0) \)[/tex].
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.