Rational inequality algebraically

1. Using algebra, solve the inequality

$$\frac { 1 } { x + 2 } > 2 x + 3$$

2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 } { x - 3 } \leqslant x + 2$$

1. Using algebra, find the set of values of \(x\) for which

$$\frac { x } { x + 2 } < \frac { 2 } { x + 5 }$$

1. Use algebra to find the set of values of \(x\) for which

$$\frac { x - 4 } { ( x + 3 ) } \leqslant \frac { 5 } { x ( x + 3 ) }$$

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

\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where \(k\) is a positive constant,

1. show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where \(p , q\) and \(r\) are expressions in terms of \(k\) to be determined.
2. Hence, or otherwise, determine the values for \(x\) for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$

1. 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 $$\frac { x + 1 } { ( x - 3 ) ( x + 2 ) } \leqslant 1 - \frac { 2 } { x - 3 }$$

5. Using algebra, find the set of values of \(x\) for which \(2 x - 5 > \frac { 3 } { x }\).

5. Solve the inequality \(\frac { 1 } { 2 x + 1 } > \frac { x } { 3 x - 2 }\).

5. Find the set of values of \(x\) for which $$\frac { x + 1 } { 2 x - 3 } < \frac { 1 } { x - 3 }$$

1. Find the set of values of \(x\) for which

$$\frac { 3 } { x + 3 } > \frac { x - 4 } { x }$$

2. Use algebra to find the set of values of \(x\) for which $$\frac { 6 x } { 3 - x } > \frac { 1 } { x + 1 }$$

2. Using algebra, find the set of values of \(x\) for which $$3 x - 5 < \frac { 2 } { x }$$

1. Use algebra to find the set of values of \(x\) for which

$$\frac { x } { x + 1 } < \frac { 2 } { x + 2 }$$

2. Use algebra to find the set of values of \(x\) for which $$\frac { x - 2 } { 2 ( x + 2 ) } \leqslant \frac { 12 } { x ( x + 2 ) }$$ "

1. Find the set of values of \(x\) for which

$$\frac { x } { x - 3 } > \frac { 1 } { x - 2 }$$

2. Use algebra to determine the set of values of \(x\) for which $$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$ (Solutions relying entirely on graphical methods are not acceptable.)
(8)

1. Use algebra to find the set of values of \(x\) for which

$$\frac { 1 } { x - 2 } > \frac { 2 } { x }$$

4

1. Show that \(x ^ { 2 } - x + 2 > 0\) for all real \(x\).
2. Solve the inequality \(\frac { 2 x } { x ^ { 2 } - x + 2 } > x\).

3

1. Sketch the graph of \(y = \frac { 2 } { x + 4 }\).
2. Solve the inequality $$\frac { 2 } { x + 4 } \leqslant x + 3$$ showing your working clearly.

4 Solve the inequality \(\frac { 5 x } { x ^ { 2 } + 4 } < x\).

4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).

1. (a) Sketch the curve with equation

$$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$

4 A curve has equation $$y = \frac { 6 x } { x - 1 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve and the two asymptotes.
3. Solve the inequality $$\frac { 6 x } { x - 1 } < 3$$

1. Use algebra to find the values of \(x\) for which

$$\frac { x } { x ^ { 2 } - 2 x - 3 } \leqslant \frac { 1 } { x + 3 }$$

1. A student was set the following problem.

Use algebra to find the set of values of \(x\) for which $$\frac { x } { x - 24 } > \frac { 1 } { x + 11 }$$ The student's attempt at a solution is written below. $$\begin{gathered} x ( x - 24 ) ( x + 11 ) ^ { 2 } > ( x + 11 ) ( x - 24 ) ^ { 2 } \\ x ( x - 24 ) ( x + 11 ) ^ { 2 } - ( x + 11 ) ( x - 24 ) ^ { 2 } > 0 \\ ( x - 24 ) ( x + 11 ) [ x ( x + 11 ) - x - 24 ] > 0 \\ ( x - 24 ) ( x + 11 ) \left[ x ^ { 2 } + 10 x - 24 \right] > 0 \\ ( x - 24 ) ( x + 11 ) ( x + 12 ) ( x - 2 ) > 0 \\ x = 24 , x = - 11 , x = - 12 , x = 2 \\ \{ x \in \mathbb { R } : - 12 < x < - 11 \} \cup \{ x \in \mathbb { R } : 2 < x < 24 \} \end{gathered}$$ Line 3 There are errors in the student's solution.

1. Identify the error made
   1. in line 3
   2. in line 7
2. Find a correct solution to this problem.