Question

which inequality does the given graph represent? a ( y > x - 3 ) b ( y geq -x - \frac{1}{3} ) c ( y < -x + 2 ) d ( y > -x - 3 )

Explanation:

Step1: Determine the slope and y-intercept of the boundary line

The boundary line is a straight line. Let's find two points on the line. From the graph, we can see that the line passes through \((-3, 0)\) and \((0, -3)\) (approximate points, we can calculate the slope \(m\) using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So \(m=\frac{-3 - 0}{0 - (-3)}=\frac{-3}{3}=- 1\). The y-intercept \(b\) is the value of \(y\) when \(x = 0\), so from the point \((0,-3)\), \(b=-3\). So the equation of the boundary line in slope - intercept form \(y=mx + b\) is \(y=-x - 3\).

Step2: Determine the inequality symbol

The line is dashed (since the inequality is strict, either \(>\) or \(<\), not \(\geq\) or \(\leq\)). Now, we need to determine which side of the line is shaded. Let's pick a test point in the shaded region. Let's take the point \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>-x - 3\): \(0>-0 - 3\), which simplifies to \(0>-3\), which is true. Now check the other options:

• For option A: \(y>x - 3\), substitute \((0,0)\): \(0>0 - 3\) (true), but the slope of the line in option A is \(1\), while our boundary line has a slope of \(-1\), so A is incorrect.
• For option B: The line is \(y\geq - x-\frac{1}{3}\), but our boundary line has a y-intercept of \(-3\) not \(-\frac{1}{3}\), and the line is dashed (not solid), so B is incorrect.
• For option C: \(y<-x + 2\), substitute \((0,0)\): \(0<-0 + 2\) (true), but the y-intercept of the line in option C is \(2\), while our boundary line has a y-intercept of \(-3\), so C is incorrect.
• For option D: \(y>-x - 3\), we already saw that the boundary line equation is \(y=-x - 3\), the line is dashed, and the test point \((0,0)\) satisfies \(0>-0 - 3\).

Answer:

D. \(y > -x - 3\)