Question

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Explanation:

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

For a point \((x,y)\), substitute into the inequality.

For point a \((2,4)\):

Left side \( y = 4 \), right side \( -\frac{1}{2}x=-\frac{1}{2}\times2 = - 1 \). Since \( 4\geq - 1 \), the point satisfies \( y\geq-\frac{1}{2}x \).

For point b \((-3,5)\):

Left side \( y = 5 \), right side \( -\frac{1}{2}x=-\frac{1}{2}\times(-3)=\frac{3}{2} = 1.5 \). Since \( 5\geq1.5 \), the point satisfies \( y\geq-\frac{1}{2}x \).

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

Simplify the inequality: divide both sides by 2, we get \( x + 2y<10 \), or \( 2y<-x + 10 \), \( y<-\frac{1}{2}x + 5 \)

For point a \((2,4)\):

Substitute \( x = 2,y = 4 \) into \( 2x + 4y \): \( 2\times2+4\times4=4 + 16=20 \). But the inequality is \( 2x + 4y<20 \), and \( 20\) is not less than \( 20 \), so point a does not satisfy \( 2x + 4y<20 \).

For point b \((-3,5)\):

Substitute \( x=-3,y = 5 \) into \( 2x + 4y \): \( 2\times(-3)+4\times5=-6 + 20 = 14 \). Since \( 14<20 \), the point satisfies \( 2x + 4y<20 \).

Answer:

a. \((2,4)\) does not satisfy the system of inequalities.
b. \((-3,5)\) satisfies the system of inequalities.