Solving Exponential Inequalities Quiz

Exponential Function Properties

• For an exponential function with base b greater than 1, if x1 is greater than x2, then b^x1 will be greater than b^x2.

Solving Exponential Inequalities

• The key to solving exponential inequalities lies in understanding the behavior of exponential functions with respect to their base.
  • If the base is greater than 1, the function increases as the exponent increases.
  • If the base is between 0 and 1, the function decreases as the exponent increases.

Solving Linear Inequalities

• The solution set for the inequality 2x + 9 ≤ 4x + 1 is x ≥ 4.
  • To solve, isolate x on one side of the inequality.

Solving Exponential Inequalities with Fractional Bases

• The solution set for the inequality (0.5)^1/343 (x - 5) ≤ 3(x + 8) is x ≥ -2.94.
  • Simplify the fractional base by converting it to a power of 1/2.
  • Isolate x on one side of the inequality and remember to flip the inequality sign if multiplying or dividing both sides by a negative number.

Solving Exponential Inequalities with Fractional Exponents

• The solution set for the inequality 3x > (0.25)^-x-2 is x > -2.
  • Rewrite 0.25 as 1/4 and express the power as a positive exponent.
  • Isolate x on one side of the inequality and remember to flip the inequality sign if multiplying or dividing both sides by a negative number.

Solving Rational Inequalities

• The solution set for the inequality 125 / (2x + 8) ≤ 512 / (x + 5) is x < -5 or -5<x<11/2.
  • Subtract the right side of the inequality from the left side and simplify to a single fraction.
  • Find the values of x that make the numerator or denominator equal to zero.
  • Test each interval to determine if the inequality holds true.