To determine the true statements about the linear inequality [tex]\( y > \frac{3}{4}x - 2 \)[/tex], we need to analyze the characteristics of the inequality and the properties of its graph.
1. The slope of the line is -2:
- Consider the inequality [tex]\( y > \frac{3}{4}x - 2 \)[/tex]. The standard 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 this case, [tex]\( \frac{3}{4} \)[/tex] is the coefficient of [tex]\( x \)[/tex], which means the slope of the line is [tex]\( \frac{3}{4} \)[/tex], not [tex]\(-2\)[/tex].
- Therefore, this statement is false.
2. The graph of [tex]\( y > \frac{3}{4}x - 2 \)[/tex] is a dashed line:
- The inequality [tex]\( y > \frac{3}{4}x - 2 \)[/tex] is strict (it does not include equality). Inequalities of the form [tex]\( y > \)[/tex] or [tex]\( y < \)[/tex] are represented by dashed lines because the line itself is not part of the solution.
- Therefore, this statement is true.
3. The area below the line is shaded:
- The inequality [tex]\( y > \frac{3}{4}x - 2 \)[/tex] means that [tex]\( y \)[/tex] is greater than [tex]\( \frac{3}{4}x - 2 \)[/tex], so we consider the region above the line where the [tex]\( y \)[/tex]-values are higher than those given by the line.
- Therefore, this statement is false.
4. One solution to the inequality is [tex]\((0,0)\)[/tex]:
- To verify if [tex]\((0,0)\)[/tex] is a solution, we substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[
0 > \frac{3}{4} \cdot 0 - 2 \quad \Rightarrow \quad 0 > -2
\][/tex]
- The inequality [tex]\( 0 > -2 \)[/tex] is true.
- Therefore, this statement is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\((0,-2)\)[/tex]:
- The y-intercept of a line is found by setting [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = \frac{3}{4}x - 2 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex], we get [tex]\( y = -2 \)[/tex], so the graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\((0, -2)\)[/tex].
- Therefore, this statement is true.
Based on the analysis:
- 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 three true statements are:
1. The graph of [tex]\( y > \frac{3}{4}x - 2 \)[/tex] is a dashed line.
2. One solution to the inequality is [tex]\((0,0)\)[/tex].
3. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\((0,-2)\)[/tex].
