Graph the system of inequalities. Then use your graph to identify the point that represents a solution to the system. x > ¨C2 y ¡Ü 2x + 7
A. (¨C1, 6)
B. (1, 11)
C. (¨C1, 4)
D. (¨C3, ¨C1)
To graph the system of inequalities, let's start with the first inequality: x > -2.
We can represent this inequality by drawing a dashed vertical line at x = -2, since x is greater than -2 but not including -2.
Next, let's graph the second inequality: y ≤ 2x + 7.
To do this, we start by graphing the line y = 2x + 7. We can start with the y-intercept, which is 7, and then use the slope of 2 to find additional points on the line. We can draw a solid line to represent this equation.
Now, let's shade the region that satisfies both inequalities. Since the second inequality is y ≤ 2x + 7, which means y is less than or equal to 2x + 7, we shade below the line so that the point (x, y) is below the line y = 2x + 7.
Finally, we can identify the point that represents a solution to the system by looking for the intersection point of the shaded region and the dashed line x = -2. From the given options, we can see that the point (-1, 4) is the only one that falls within the shaded region and satisfies the system of inequalities.
Therefore, the point that represents a solution to the system is C. (-1, 4).
To graph the system of inequalities, we will plot the individual inequality equations on the coordinate plane and find the overlapping region.
1. Let's start with the first inequality: x > -2. Since this is a "greater than" inequality, we need to use a dashed line to indicate that the solution does not include the boundary line itself. To graph this, draw a dashed vertical line at x = -2.
2. Now, let's move on to the second inequality: y <= 2x + 7. This is a "less than or equal to" inequality, so we need to use a solid line to indicate that the solution includes the boundary line. To graph this, start by finding its slope-intercept form (y = mx + b). Rearranging the inequality, we get y - 2x <= 7. This means the line has a slope of 2 and a y-intercept of 7. Plot the y-intercept (0,7) and use the slope to find another point, for example, by going up 2 units and right 1 unit from the y-intercept. Connect these two points with a solid line because the inequality is "less than or equal to."
3. Now that we have both inequalities graphed on the same coordinate plane, we need to find the overlapping region. This is the area where both inequalities are true. Shade the region below or above the respective lines, depending on the direction of the inequality symbol for each equation.
4. Finally, we need to identify the point that represents a solution to the system. Look for the intersection point of the shaded region. This is the precise point where both inequalities are simultaneously satisfied.
From the given answer choices, we can see that the point (−1, 6) lies in the shaded region and is also the intersection point of the two lines. Therefore, the correct answer is:
A. (−1, 6)