Question

question 2 1 point when graphing the following system of inequalities, which of the following statements correctly describes how the graph should look? ( y geq -x - 1 ) ( y < \frac{1}{3}x + 3 ) a solid line shaded below the line with a slope of -1 and a y-intercept of 3. a dashed line shaded above the line with a slope of ( \frac{1}{3} ) and a y-intercept of -1. a solid line shaded above the line with a slope of ( \frac{1}{3} ) and a y-intercept of 3. a dashed line shaded below the line with a slope of -1 and a y-intercept of 3. a solid line shaded above the line with a slope of -1 and a y-intercept of 3. a dashed line shaded below the line with a slope of ( \frac{1}{3} ) and a y-intercept of 3.

Explanation:

Step1: Analyze \( y \geq -x - 1 \)

The inequality \( y \geq -x - 1 \) is in slope - intercept form \( y=mx + b \), where \( m=- 1 \) (slope) and \( b = - 1 \) (y - intercept). Since the inequality is \( \geq \), we use a solid line (because the points on the line are included in the solution set) and shade above the line (because for a point \((x,y)\) to satisfy \( y\geq -x - 1\), \( y\) values are greater than or equal to the values on the line \( y=-x - 1\)).

Step2: Analyze \( y<\frac{1}{3}x + 3 \)

The inequality \( y<\frac{1}{3}x + 3 \) is in slope - intercept form \( y = mx + b \), where \( m=\frac{1}{3}\) (slope) and \( b = 3 \) (y - intercept). Since the inequality is \( < \), we use a dashed line (because the points on the line are not included in the solution set) and shade below the line (because for a point \((x,y)\) to satisfy \( y<\frac{1}{3}x + 3\), \( y\) values are less than the values on the line \( y = \frac{1}{3}x+3\)).

Now let's check the options:

• Option 1: Says "A solid line shaded below the line with a slope of - 1. A dashed line shaded above the line with a slope of \(\frac{1}{3}\) and a y - intercept of 3". The shading for \( y\geq -x - 1\) should be above, not below. So this is incorrect.
• Option 2: "A solid line shaded above the line with a slope of \(\frac{1}{3}\) and a y - intercept of 3. A dashed line shaded below the line with a slope of - 1 and a y - intercept of - 1". The slopes and intercepts are mixed up. The line \( y=-x - 1\) has slope - 1 and \( y = \frac{1}{3}x + 3\) has slope \(\frac{1}{3}\). So this is incorrect.
• Option 3: "A dashed line shaded above the line with a slope of - 1 and a y - intercept of - 1. A solid line shaded below the line with a slope of \(\frac{1}{3}\) and a y - intercept of 3". The line \( y=-x - 1\) should be solid (because of \(\geq\)) and \( y<\frac{1}{3}x + 3\) should be dashed. Also, the shading for \( y=-x - 1\) is above and for \( y=\frac{1}{3}x + 3\) is below, but the line types are wrong. So this is incorrect.
• Option 4: "A solid line shaded above the line with a slope of - 1 and a y - intercept of - 1. A dashed line shaded below the line with a slope of \(\frac{1}{3}\) and a y - intercept of 3". For \( y\geq -x - 1\), we have a solid line (slope - 1, y - intercept - 1) and shade above. For \( y<\frac{1}{3}x + 3\), we have a dashed line (slope \(\frac{1}{3}\), y - intercept 3) and shade below. This matches our analysis.

Answer:

The correct option is the one that describes "A solid line shaded above the line with a slope of - 1 and a y - intercept of - 1. A dashed line shaded below the line with a slope of \(\frac{1}{3}\) and a y - intercept of 3" (the fourth option in the list of options provided in the problem).