Edexcel IAL Mathematics FP2.1.1 algebraic inequalities

Practise solving algebraic inequalities and inequations, including modulus and rational forms, using algebra rather than graphical methods.

Exam points

split modulus inequalities into cases and combine the resulting intervals for x
solve rational inequalities by finding critical values and testing valid intervals
state excluded values clearly when an inequation contains algebraic fractions

[Maximum number: 6]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use algebra to determine the values of x for which

\frac{x+1}{(x-3)(x+2)}\le 1-\frac{2}{x-3}

Critical values from the denominator are x=-2 and x=3.

Equating the two sides:

\frac{x+1}{(x-3)(x+2)}=1-\frac{2}{x-3}

x+1=x^2-x-6-2(x+2)

x^2-4x-11=0.

x=\frac{4\pm\sqrt{16+44}}{2}=2\pm\sqrt{15}

Therefore

x<-2,\qquad 2-\sqrt{15}\le x<3,\qquad x\ge 2+\sqrt{15}

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FP2.1.1 - Manipulation and solution question 1