Answer:
1. [tex] x \ge -4 [/tex]
2. [tex] x < -2 [/tex]
Step-by-step explanation:
1. [tex] 10^{2x - 4} \le 1000^x [/tex]
Make both sides of the same base
[tex] 10^{2x - 4} \le (10^3)^x [/tex]
[tex] 10^{2x - 4} \le 10^{3x} [/tex]
Both bases will cancel each other
[tex] 2x - 4 \le 3x [/tex]
Subtract 2x from each side
[tex] - 4 \le 3x - 2x [/tex]
[tex] -4 \le x [/tex]
Rewrite
[tex] x \ge -4 [/tex]
2. [tex] (\frac{1}{5})^{3x + 10} > (\frac{1}{25}^{x + 4} [/tex]
Apply the inverse law of exponents
[tex] (5^{-1})^{3x + 10} > (25^{-1})^{x + 4} [/tex]
[tex] (5^{-1})^{3x + 10} > (5^{-2})^{x + 4} [/tex]
[tex] 5^{-3x - 10} > 5^{-2x - 8} [/tex]
Both bases will cancel each other
[tex] -3x - 10 > -2x - 8 [/tex]
Collect like terms
[tex] -3x + 2x > 10 - 8 [/tex]
[tex] -x > 2 [/tex]
Divide both sides by -1.
(Note: since we are dividing by a negative number, the inequality sign would change)
[tex] x < -2 [/tex]
