Question

look at the factorization shown below.
$0 < -100x^2 + 350x - 150$
$0 < -50(2x^2 - 7x + 3)$
$0 < -50(2x - 1)(x - 3)$
select the range that mario can choose from to price his muffins and make a positive profit.
$\bigcirc$ $x < \\$0.50$
$\bigcirc$ $\\$0.50 < x < \\$3.00$
$\bigcirc$ $\\$0.50 < x$ or $x > \\$3.00$
$\bigcirc$ $x > \\$3.00$

Explanation:

Step1: Simplify the inequality

Divide both sides by $-50$, reverse inequality sign:
$0 > 2x^2 -7x +3$
Which is equivalent to $2x^2 -7x +3 < 0$

Step2: Find critical points

Set factored terms to 0:
$2x-1=0 \implies x=\frac{1}{2}=0.50$
$x-3=0 \implies x=3.00$

Step3: Test intervals for sign

Test $x<0.50$ (e.g., $x=0$): $2(0)^2-7(0)+3=3>0$ (not valid)
Test $0.50Test $x>3.00$ (e.g., $x=4$): $2(16)-7(4)+3=32-28+3=7>0$ (not valid)

Answer:

$\$0.50 < x < \$3.00$