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Test: Rational, exponential and logarithmic inequalities

This test develops methods for rational, exponential and logarithmic inequalities. Students use the interval method, account for monotonicity of exponential or logarithmic functions and enforce all domain restrictions.

Question qty: 10
Duration in minutes: 20
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Sample questions

  • Solve the system a < x < b and x ≥ c.
  • Solve (x − (a))/(x − (b)) < 0.
  • For 0 < a < 1, solve a^x > a^r.
  • For 0 < a < 1, solve log_a(x) > log_a(v).
  • Solve (x − (a))/(x − (b)) > 0.
  • For a > 1, solve log_a(x) > log_a(v).
  • For a > 1, solve a^x > a^r.
  • Which points are important when applying the interval method to a rational inequality?

Rational, exponential and logarithmic inequalities

Solve a rational inequality with a sign chart based on zeros of its numerator and denominator. Denominator zeros are never included, even with ≤ or ≥.

Monotonicity and domains

For an exponential base a > 1, the direction of an exponent inequality is preserved; for 0 < a < 1, it reverses. Logarithmic inequalities follow the same increasing or decreasing behaviour, but every argument must first be positive. Give the final answer as the intersection of all permitted intervals.

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