1.02g Inequalities: linear and quadratic in single variable

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

4 Solve the inequality \(5 - 2 x < 0\).

5 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$

4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).

6 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).

3

1. Solve the inequality \(\frac { 1 - 2 x } { 4 } > 3\).
2. Simplify \(\left( 5 c ^ { 2 } d \right) ^ { 3 } \times \frac { 2 c ^ { 4 } } { d ^ { 5 } }\).

8 Fig. 8 shows part of the graph of \(y = \frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) }\). Two sections of the graph have been omitted. \begin{figure}[h] \end{figure}

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Copy Fig. 8 and draw in the two missing sections.
4. Solve the inequality \(\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) } \leqslant 0\).

7 A curve has equation \(y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }\).

1. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your answers.
3. Sketch the curve.
4. Solve the inequality \(\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0\).

7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the coordinates of the two points where the curve crosses the axes.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
4. On the copy of Fig. 7, sketch the rest of the curve.
5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).

7 A curve has equation \(y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }\).

1. Find the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or from below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.
5. Solve the inequality \(\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2\).

4

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

7 Fig. 7 shows a sketch of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\). \begin{figure}[h] \end{figure}

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes. Hence write down the solution of the inequality \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } > 0\).
2. The equation \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } = k\) has no real solutions. Show that \(- 1 < k < - \frac { 1 } { 9 }\). Relate this result to the graph of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).

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.

7 Fig. 7 shows an incomplete sketch of \(y = \frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) }\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the points where the curve cuts the axes.
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of \(x\), justifying your answer. Copy and complete the sketch.
4. Solve the inequality \(\frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) } < 2\).

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

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

7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.

1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
3. Sketch the curve.
4. Find the set of values of \(x\) for which \(y > 0\).

7 A curve has equation \(\mathrm { y } = \frac { ( 3 \mathrm { x } + 2 ) ( \mathrm { x } - 3 ) } { ( \mathrm { x } - 2 ) ( \mathrm { x } + 1 ) }\).

1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes.
2. Sketch the curve, justifying how it approaches the horizontal asymptote.
3. Find the set of values of \(x\) for which \(y \geqslant 3\).

8 A curve has equation \(y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }\).

1. Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.
2. State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
3. Sketch the curve.
4. Solve the inequality \(\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0\).

8 Four distinct positive integers are \(( 3 x - 5 ) , ( 2 x + 3 ) , ( x + 1 )\) and \(( 4 x - 13 )\).

1. Explain why \(x \geqslant 4\).
2. The four integers are to be sorted into ascending order using a bubble sort. The original list is \(\begin{array} { c c c c } ( 3 x - 5 ) & ( 2 x + 3 ) & ( x + 1 ) & ( 4 x - 13 ) \end{array}\) After the first pass, the list is \(( 3 x - 5 ) \quad ( x + 1 ) \quad ( 4 x - 13 ) \quad ( 2 x + 3 )\) After the second pass, the list is \(( x + 1 )\) \(( 4 x - 13 )\) \(( 3 x - 5 )\) \(( 2 x + 3 )\) After the third pass, the list is \(( 4 x - 13 ) \quad ( x + 1 )\) \(( 3 x - 5 )\) ( \(2 x + 3\) )
   1. By considering the list after the first pass, write down three inequalities in terms of \(x\).
   2. By considering the list after the second pass, write down two further inequalities in terms of \(x\).
   3. By considering the list after the third pass, write down one further inequality in terms of \(x\).
3. Hence, by considering the results above, find the value of \(x\).
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2 Five different integers are to be sorted into ascending order.

1. A bubble sort is to be used on the list of numbers \(\quad 6 \quad 4 \quad 5 \quad x \quad 2 \quad 11\).
   1. After the first pass, the list of numbers becomes $$\begin{array} { l l l l l } 4 & x & 2 & 6 & 11 \end{array}$$ Write down an inequality that \(x\) must satisfy.
   2. After the second pass, the list becomes $$\begin{array} { l l l l l } x & 2 & 4 & 6 & 11 \end{array}$$ Write down a new inequality that \(x\) must satisfy.
2. The five integers are now written in a different order. A shuttle sort is to be used on the list of numbers \(\quad \begin{array} { l l l l l } 11 & x & 2 & 4 & 6 . \end{array}\)
   1. After the first pass, the list of numbers becomes $$\begin{array} { l l l l l } x & 11 & 2 & 4 & 6 \end{array}$$ Write down an inequality that \(x\) must satisfy.
   2. After the second pass, the list becomes $$\begin{array} { l l l l l } 2 & x & 11 & 4 & 6 \end{array}$$ Write down a further inequality that \(x\) must satisfy.
3. Use your answers from parts (a) and (b) to write down the value of \(x\).

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).

1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.

3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
4. Comment briefly on how realistic model B may be in the following cases.

1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
   (b) find \(| \overrightarrow { A B } |\).

Give your answer as a simplified surd.

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 $$\frac { 16 } { x } \leqslant 2$$