NSW Y11 Maths - Extension 1 Functions Inequalities -X In Denominator

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For inequalities such as \(\dfrac{x}{{\left( {x - a} \right)}} > b\), both sides of the inequality must be multiplied by \({\left( {x - a} \right)^2}\) as this will not affect the inequality sign and at the same time eliminate the term in the denominator.

It is important not to remove brackets but to transpose all the terms to one side of the inequality and then factorise. The critical points can then be determined, and the inequality solved.

NSW Syllabus Reference: ME-F1.2 Inequalities. This will require student to 

• solve quadratic inequalities using both algebraic and graphical techniques
• solve inequalities involving rational expressions, including those with the unknown in the denominator
• solve absolute value inequalities of the form \(|ax+b| \ge k,\, |ax+b| \le k,\, |ax+b| <k\) and \(|ax+b| > k\)

Ref: https://educationstandards.nsw.edu.au/