Answer:
Option 3: 4x - 2y < -8
Step-by-step explanation:
Given the graph of a linear inequality with a dashed line as its boundary line:
Quick Note:
To start with, since the graph of the line is dashed, this means that the endpoints are not included as solutions.
- Hence, we can easily rule out Option 4: 2x + 4y ≥ -16, since the solid boundary lines are used to represent linear inequalities with "≥" or "≤" symbols.
Solutions:
It will help to choose a test point that is not on the line, and substitute its values into the given inequality statements. Test points are used to verify whether it will satisfy the given inequality statement.
Let's use the point of origin as our test point, (0, 0). Substitute its values into the linear inequality statements in Options 1 through 3:
Option 1: 4x - 3y > -12
4x - 3y > -12
4(0) - 3(0) > -12
0 - 0 > -12
0 > -12 (True statement). Hence, the half-plane region that contains the test point must be shaded.
Since the point of origin is not a part of the shaded region in the given graph, then it means that the linear inequality in Option 1 is not the correct answer.
Option 2: x + 4y > 4
x + 4y > 4
(0) + 4(0) > 4
0 + 0 > 4
0 > 4 (False statement). The half-plane region that doesn't contain the test point must be shaded. Hence, Option 2 may be a valid solution.
We must transform the given inequality statement into its slope-intercept form, to see whether its slope matches the graph:
x + 4y > 4
x - x + 4y > - x + 4
4y > -x + 4
Divide both sides by 4 to isolate y:
[tex]\displaystyle\mathsf{\frac{4y}{4}>\:\frac{-x\:+\:4}{4} }[/tex]
y > -¼x + 1 ⇒ This is the slope-intercept form, where it has a negative slope of m = -¼. Hence, Option 2 is not a valid answer, as the graph has a positive slope.
Option 3: 4x - 2y < - 8
4x - 2y < - 8
4(0) - 2(0) < - 8
0 - 0 < - 8
0 < -8 (False statement). Thus, the half-plane region that doesn't contain the test point must be shaded.
We must transform the given inequality statement into its slope-intercept form, to see whether its slope matches the graph:
4x - 2y < - 8
4x - 4x - 2y < -4x - 8
-2y < -4x - 8
Divide both sides by -2 to isolate y:
[tex]\displaystyle\mathsf{\frac{-2y}{-2}>\:\frac{-4x\:-\:8}{-2} }[/tex]
y < 2x + 4 ⇒ This linear inequality statement in slope-intercept form matches the given graph, as its line has a positive slope of 2.
Therefore, the correct answer is Option 3: 4x - 2y < -8.
