Let's analyze the given linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex].
1. The slope of the line is -2.
- The slope of a line in the equation [tex]\( y = mx + b \)[/tex] is given by the coefficient of [tex]\( x \)[/tex], which in this case is [tex]\( \frac{3}{4} \)[/tex]. Therefore, 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.
- When graphing a linear inequality that uses the "greater than" (>) symbol, the boundary line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is drawn as a dashed line to indicate that points on the line are not included in the solution. Thus, this statement is true.
3. The area below the line is shaded.
- For [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the shaded region consists of all points above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] because we are looking for the [tex]\( y \)[/tex]-values that are greater than those on 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]\( y > \frac{3}{4} x - 2 \)[/tex]:
[tex]\[ 0 > \frac{3}{4}(0) - 2 \][/tex]
[tex]\[ 0 > -2 \][/tex]
This inequality is true. Therefore, [tex]\( (0, 0) \)[/tex] is indeed a solution of the inequality. This statement is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, -2) \)[/tex].
- To find the [tex]\( y \)[/tex]-intercept of the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex], we set [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{3}{4}(0) - 2 \][/tex]
[tex]\[ y = -2 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the graph is at [tex]\( (0, -2) \)[/tex]. This statement is true.
To summarize, the true statements about the 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].
