Solve:
a) 1/(2^x) = 1/(x+2)
b) 1/(2^x) > 1/(x^2)
x> (7i + 1) / 2
for the first one, cross-multiply
2x^2 = x+2
2x^2 - x - 2 = 0
use the quadratic formula to solve.
For the second, let's see if the corresponding functions intersect, let
1/(2^x) = 1/(x^2)
2x^2 = x^2
true only for x = 0
but x cannot be zero in the original,
so they do not intersect.
clearly for │x│>1
2x^2 > x^2, so
1/(2x^2) < 1/x^2 is false
e.g.
let x=5
1/50 > 1/25 ?? ---> false
but for │x│ < 1 it is true
(the denominator of the first is bigger the second, so the first fraction is smaller)
e.g.
x = .7
1/((2)(.49)) < 1/(.49) ?? ---> true
a) To solve the equation 1/(2^x) = 1/(x+2), we can follow these steps:
Step 1: Cross-multiply the equation.
1 * (x + 2) = 1 * (2^x)
x + 2 = 2^x
Step 2: Rewrite the equation as a power of 2.
2^x = x + 2
Step 3: Rearrange the equation to isolate one side.
2^x - x = 2
Step 4: Solve the equation using numerical methods or with the help of technology. Since this equation does not have a straightforward algebraic solution, we can use numerical methods to approximate the solution. For example, you can graph the functions y = 2^x - x and y = 2 to find their intersection point.
b) To solve the inequality 1/(2^x) > 1/(x^2), we can follow these steps:
Step 1: Cross-multiply the inequality, but be careful with the direction of the inequality sign.
1 * (x^2) > 1 * (2^x)
x^2 > 2^x
Step 2: Rewrite the inequality as a power of 2.
2^x < x^2
Step 3: Rearrange the inequality to isolate one side.
x^2 - 2^x > 0
Step 4: Analyze the inequality by studying the behavior of each side. We can make a table or use technology to plot the graph of the functions y = x^2 - 2^x and y = 0. By examining where the function is positive or negative, we can find the solution to the inequality.