Answer:
The amount invested at Investment A must be greater than or equal to $2,750
The amount invested at Investment B must be less than or equal to $2,250
Step-by-step explanation:
Let
x -----> the amount invested at Investment A yields 5%
y -----> the amount invested at Investment B yields 8%
we know that
[tex]x+y=5,000[/tex]
[tex]x=5,000-y[/tex] -----> equation A
[tex]x \geq 0.25(5,000)[/tex]
[tex]x \geq \$1,250[/tex] -----> inequality B
[tex]y \leq 0.50(5,000)[/tex]
[tex]y \leq \$2,250[/tex] -----> inequality C
[tex]x \geq \frac{1}{2}y[/tex] -----> inequality D
Substitute equation A in the inequality D and solve for y
[tex]5,000-y \geq \frac{1}{2}y[/tex]
Multiply by 2 both sides
[tex]10,000-2y \geq y[/tex]
Multiply by -1 both sides
[tex]-10,000+2y \leq -y[/tex]
Adds y both sides
[tex]-10,000+2y+y \leq 0[/tex]
[tex]-10,000+3y \leq 0[/tex]
adds 10,000 both sides
[tex]3y \leq 10,000[/tex]
Divide by 3 both sides
[tex]y \leq \$3,333.33[/tex] -----> inequality E
therefore
Solve for y
we have
[tex]y \leq \$2,250[/tex] -----> inequality C
[tex]y \leq \$3,333.33[/tex] -----> inequality E
The solution of inequality C and inequality E is
[tex]y \leq \$2,250[/tex]
For y=2,250
x=5,000-y ----> x=5,000-2,250=2,750
so
[tex]x \geq \$2,750[/tex] -----> inequality F
Solve for x
we have
[tex]x \geq \$1,250[/tex] -----> inequality B
[tex]x \geq \$2,750[/tex] -----> inequality F
The solution of inequality B and inequality F is
[tex]x \geq \$2,750[/tex]
therefore
The amount invested at Investment A must be greater than or equal to $2,750
The amount invested at Investment B must be less than or equal to $2,250
