Let's go through all the options one at a time and look for the correct ones
<u>Option 1: The slope of one boundary line is 2</u>
We have 2 equations of lines, where the coefficients of are 3 and -1 respectively
because the coefficient of x denotes the slope of a line, we know that the lines have the slope 3 and -1, not 2
Hence, this option is Incorrect
<u>Option 2: Both boundary lines are solid</u>
In order for the boundary lines to be solid, the inequality must have an 'equal to', like ≤ (less than or equal to) or ≥ (greater than or equal to)
we can see that that's the case in our case and hence, this option is Correct
<u>Option 3: A solution to the system is (1, 3)</u>
To confirm this, we'll plug these coordinates into the given inequalities and see if it stands correct
y ≤ 3x + 1
3 ≤ 3(1) + 1
3 ≤ 4 which is correct because 3 is less than 4
Second equation:
y ≥ 2 - x
3 ≥ 2 - 1
3 ≥ 1
Which is also true because 3 is greater than 1
Now, we can say that (1 , 3) is a solution to the system because it satisfies both the equations and is Correct
<u>Option 4: Both inequalities are shaded below the boundary lines</u>
For an inequality to be shaded below the boundary line, it must have the ≤ inequality (in case of solid line) and < inequality (in case of dotted line)
because the second inequality listed includes the ≥ inequality, which was not mentioned above, it won't be shaded below
another way to think about it is that any 'greater than' inequality will shade everything above the line and the 'lesser than' inequality will shade below the line
which means that this option is Incorrect
<u>Option 5: The boundary lines intersect</u>
In order for the boundary lines to intersect, they must have have different slopes.
as we mentioned in the explanation of the first option, that the slopes of the lines is 3 and -1, which are different slopes
Therefore, this option is Correct
