Let's examine each statement about the linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] and determine which ones are true.
1. The slope of the line is -2.
- This statement is about the slope of the line represented by the inequality. The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] has the form [tex]\( y > mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. In this case, [tex]\( m \)[/tex] is [tex]\( \frac{3}{4} \)[/tex]. Thus, the slope of the line is [tex]\( \frac{3}{4} \)[/tex], not -2. This statement is false.
2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- In an inequality of the form [tex]\( y > mx + b \)[/tex], the line is dashed because the points on the line itself do not satisfy the inequality (since it is a strict inequality, not [tex]\( \geq \)[/tex]). This statement is true.
3. The area below the line is shaded.
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] specifies that [tex]\( y \)[/tex] is greater than [tex]\( \frac{3}{4} x - 2 \)[/tex]. This means that the region above the line is shaded, not the area below. This statement is false.
4. One solution to the inequality is [tex]\((0,0)\)[/tex].
- To determine if [tex]\((0,0)\)[/tex] is a solution, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[
0 > \frac{3}{4} \cdot 0 - 2
\][/tex]
Simplifying this, we get:
[tex]\[
0 > -2
\][/tex]
This is a true statement, so [tex]\((0,0)\)[/tex] is indeed a solution to the inequality. This statement is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\((0, -2)\)[/tex].
- To find where the line intersects the [tex]\( y \)[/tex]-axis, set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3}{4} \cdot 0 - 2 = -2.
\][/tex]
Thus, the line crosses the [tex]\( y \)[/tex]-axis at the point [tex]\((0, -2)\)[/tex]. This statement is true.
Based on this analysis, the three true statements about the linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] 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].
Therefore, the correct options are (2), (4), and (5).
