Exam Questions - Modulus inequalities fractional type

Exam Questions - Modulus inequalities fractional type

Understanding the Concept of Modulus Inequalities of Fractional Type

  • A modulus inequality of fractional type involves an inequality that includes a fraction. It takes the form ** f(x) < a** or ** f(x) > a, where **f(x) is a fraction.
  • When solving such inequality, the key principle is that if f(x) is positive, ** f(x) = f(x)**
  • If f(x) is negative, ** f(x) = -f(x)**. This results in two separate inequalities to solve.
  • If the given inequality is ** f(x) < a, the solution is the intersection (AND) of the two solved inequalities, namely; **-a < f(x) < a.
  • If the given inequality is ** f(x) > a, the solutions are the union (OR) of the inequalities, namely; **f(x) > a OR f(x) < -a.

Steps in Solving Modulus Inequalities of Fractional Type

  • Start by splitting the given inequality into two inequalities based on the principles outlined above.
  • Write down the inequalities, solve each separately and graph the solution.
  • When solving a fraction inequality, try to make the denominator positive by multiplying by its square. This avoids dealing with negative denominators.
  • Find the critical points by equating your inequality to zero and solve the resulting equation to get potential boundary points.
  • Use these boundary points to divide your function into intervals. Check the sign of the function in each interval by substitifying an example point from that interval into your inequality.
  • Construct the final solution by considering the appropriate combination (union or intersection) of the solutions to each inequality based on whether the original inequality was less than or greater than.

Identifying Solutions on a Number Line

  • Solutions to modulus inequalities are often represented on a number line. Points are marked at the critical values and the valid solution intervals are shaded in.
  • If the inequality is a less than type inequality or it includes the value (≤ or ≥), then the critical values are represented with a filled circle.
  • If the inequality is a greater than type inequality and it does not include the value (< or >), then the critical values are represented with an open circle.
  • Negative solutions to the inequality will appear left of the zero point on your number line, while positive solutions appear to the right.

Remember, solving modulus inequalities of fractional type can be complex due to the presence of fractions and modulus. However, with thorough understanding of the principles and careful step-by-step working, it is manageable. Practice is key to mastering this area.