We are given a graph with lines representing inequalities. We are then asked to decipher which of the system of inequalities represents the graph.
In order to solve this question, it is easier to simply view them as normal lines with a general slope-intercept form of:
[tex]\begin{gathered} y=mx+c \\ \text{where,} \\ m=\text{slope} \\ c=y-\text{intercept (Where the graph cuts the y-axis)} \end{gathered}[/tex]
Whether or not the final expression for each line is meant to be a "greater than or equal to" or just simply "greater than" will be determined later on.
The Slope:
We need to calculate the slope (m) for both lines of the graph as this will move us 1 step closer to finding the equation of the line.
The formula for calculating the slope of a line is:
[tex]\begin{gathered} \text{Given two points,} \\ (x_1,y_1),(x_2,y_2) \\ \text{The slope of a line given these two points is:} \\ m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex]
In order to find the slope of a line, we need to find two points on that line as we can infer from the formula above.
The points chosen for both lines is given below:
As we can see from above, line AB has two coordinates: A(0, 3) and B(-3, 0) while line AC has coordinates: A(0,3) and C(1.5, 0)
With these two points, we can now find the slope of both lines.
[tex]\begin{gathered} \text{ Line AB:} \\ \text{ Using A(0, 3), B(-3, 0), we have} \\ m=\frac{0-3}{-3-0}=-\frac{3}{-3} \\ m=1 \\ \\ \text{ LIne AC:} \\ \text{Using A(0, 3), C(1.5, 0), we have} \\ m=\frac{0-3}{1.5-0}=-\frac{3}{1.5} \\ m=-2 \end{gathered}[/tex]
Thus, the slope of line AB is 1 and the slope of line AC is -2.
Since we know that lines AB and AC cross the y-axis at point y = 3, it implies that the y-intercept (c) for both lines must be y = 3 as well.
Thus, we can write equation of both lines AB and AC as:
[tex]\begin{gathered} \text{ Using y = mx + c} \\ \text{ Line AB:} \\ y=x+3 \\ \text{while,} \\ \text{ LIne AC:} \\ y=-2x+3 \end{gathered}[/tex]
Now, that we know the equations, we just have to figure out whether they are greater than or equal to or otherwise.
The way we check is to set x and y to zero and see which inequality makes sense.
After performing this operation, we find that:
[tex]\begin{gathered} y\le-2x+3 \\ y\le x+3 \\ \text{put y = 0 , x = 0} \\ 0\le3\text{ (True)} \\ 0\le3\text{ (Also true} \end{gathered}[/tex]
Therefore, the final answer is:
Option 1
