Find the Missing x!

Solve for the values of x for each of the following exponential equations and inequalities.

1. 8 ^ (2 - x) = 2

2. (1/2) * x < 1/8

3. 5 ^ x = 25 ^ (x - 2)

4. 3 ^ (x + 2) >= 27

5. 4 ^ (3x) = 8 ^ (x - 1)

Answer:Here are the solutions for each exponential equation and inequality: 1. 8^(2-x) = 2Rewrite 2 as a power of 8: 2 = 8^(1/3)Set the exponents equal: 2 - x = 1/3Solve for x: x = 5/3 2. (1/2)^x < 1/8Rewrite both sides with the same base: (1/2)^x < (1/2)^3Compare exponents: x > 3 (Remember, when dividing by a negative number, the inequality sign flips) 3. 5^x = 25^(x-2)Rewrite 25 as a power of 5: 25 = 5^2Simplify the right side: 5^x = 5^(2(x-2))Set exponents equal: x = 2(x-2)Solve for x: x = 4 4. 3^(x+2) >= 27Rewrite 27 as a power of 3: 27 = 3^3Simplify the inequality: 3^(x+2) >= 3^3Compare exponents: x + 2 >= 3Solve for x: x >= 1 5. 4^(3x) = 8^(x-1)Rewrite both sides with the same base (2): 2^(6x) = 2^(3(x-1))Set exponents equal: 6x = 3(x-1)Solve for x: x = -1

Answered by alixzamarirapacon | 2024-10-17