How do you solve #(3x +1)(5 - 10x) 0#?

Question

How do you solve #(3x +1)(5 - 10x) > 0#?

Kennedy Adams · Accepted Answer

#(3x+1)(5-10x) = -30x^2 + 5x + 5# is an inverted parabola that intersects the #x# axis at the two points, #x = -1/3# and #x = 1/2#, where its value is zero. The quadratic has a negative value when #x -> -oo# or #x -> oo#, indicating that the region in which it has a positive value is #-1/3 < x < 1/2#.

Hazel Anglin · Answer

undefined To solve $ (3x + 1)(5 - 10x) > 0 $:

1. Find the critical points where the expression equals zero:
   - $ 3x + 1 = 0 $ gives $ x = -\frac{1}{3} $
   - $ 5 - 10x = 0 $ gives $ x = \frac{1}{2} $

2. Plot these critical points on a number line.

3. Test the intervals between the critical points:
   - Test a value less than $-\frac{1}{3}$, such as $ x = -1 $, in the expression to determine if it's positive or negative.
   - Test a value between $-\frac{1}{3}$ and $\frac{1}{2}$, such as $ x = 0 $, in the expression.
   - Test a value greater than $\frac{1}{2}$, such as $ x = 1 $, in the expression.

4. Determine the intervals where the expression is positive:
   - If the expression is positive in an interval, the inequality holds true for that interval.
   - If the expression is negative in an interval, the inequality does not hold true for that interval.

5. Write the solution:
   - The solution is the union of the intervals where the expression is positive.

How do you solve #(3x +1)(5 - 10x) &gt; 0#?

How do you solve #(3x +1)(5 - 10x) > 0#?