Question

of inequalities, then determine if each orde
2
$y \geq -\frac{1}{2}x$
$2x + 4y < 20$

a. $(2, 4)$
b. $(-3, 5)$
system of inequalities to answer a-c.
inequalities represented on the graph:
airs that are solutions to the system.

Explanation:

Step1: Analyze inequality \( y \geq -\frac{1}{2}x \) for point (2,4)

Substitute \( x = 2 \), \( y = 4 \) into \( y \geq -\frac{1}{2}x \):
\( 4 \geq -\frac{1}{2}(2) \) → \( 4 \geq -1 \), which is true.

Step2: Analyze inequality \( 2x + 4y < 20 \) for point (2,4)

Substitute \( x = 2 \), \( y = 4 \) into \( 2x + 4y < 20 \):
\( 2(2) + 4(4) = 4 + 16 = 20 \). But the inequality is \( < 20 \), so \( 20 < 20 \) is false.

Step3: Analyze inequality \( y \geq -\frac{1}{2}x \) for point (-3,5)

Substitute \( x = -3 \), \( y = 5 \) into \( y \geq -\frac{1}{2}x \):
\( 5 \geq -\frac{1}{2}(-3) \) → \( 5 \geq \frac{3}{2} \), which is true.

Step4: Analyze inequality \( 2x + 4y < 20 \) for point (-3,5)

Substitute \( x = -3 \), \( y = 5 \) into \( 2x + 4y < 20 \):
\( 2(-3) + 4(5) = -6 + 20 = 14 \). Since \( 14 < 20 \), this is true.

Answer:

b. \((-3, 5)\) is a solution (as both inequalities are satisfied), while a. \((2, 4)\) is not (fails \(2x + 4y < 20\)). So the solution is b. \((-3, 5)\)