1.02g Inequalities: linear and quadratic in single variable

1 Solve the inequality \(| 2 x + 1 | < | 2 x - 5 |\).

1 Solve the inequality \(| x + 1 | < | 3 x + 5 |\).

2

1. Sketch the graph of \(y = | 2 x + 3 |\).
2. Solve the inequality \(3 x + 8 > | 2 x + 3 |\).

1

1. Sketch the graph of \(\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |\), where \(a\) is a positive constant.
2. Solve the inequality \(2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |\).

1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).

2 Solve the inequality \(| 3 x - a | > 2 | x + 2 a |\), where \(a\) is a positive constant.

2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 3\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\).

1

1. Sketch the graph of \(y = | 4 x - 2 |\).
2. Solve the inequality \(1 + 3 x < | 4 x - 2 |\).

1. The equation

$$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
Find the range of possible values of \(c\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

1. show that \(x > 1.5 \mathrm {~m}\) Given also that the area of the garden is less than \(72 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of possible values of \(x\).
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5. \begin{figure}[h] \end{figure} The share value of two companies, company \(A\) and company \(B\), has been monitored over a 15-year period. The share value \(P _ { A }\) of company \(\boldsymbol { A }\), in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. The share value \(P _ { B }\) of company \(B\), in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

1. Find the difference between the share value of company \(\boldsymbol { A }\) and the share value of company \(\boldsymbol { B }\) at the point monitoring began.
2. State the maximum share value of company \(\boldsymbol { A }\) during the 15-year period.
3. Find, using algebra and showing your working, the times during this 15-year period when the share value of company \(\boldsymbol { A }\) was greater than the share value of company \(\boldsymbol { B }\).
4. Explain why the model for company \(\boldsymbol { A }\) should not be used to predict its share value when \(t = 20\) \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Solve the inequality $$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$

7. The curve \(C\) has equation $$y = \frac { 1 } { 2 - x }$$

1. Sketch the graph of \(C\). On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line \(l\) has equation \(y = 4 x + k\), where \(k\) is a constant. Given that \(l\) meets \(C\) at two distinct points,
2. show that $$k ^ { 2 } + 16 k + 48 > 0$$
3. Hence find the range of possible values for \(k\).

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only. Given that

• \(\quad l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(\quad C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only.
Given that

• \(l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).
  

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

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.

1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
2. Hence, using algebra, find the range of possible values of \(p\)

13. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
2. Find the range of possible values for \(k\).
   

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Given \(y = \frac { x ^ { 3 } } { 3 } - 2 x ^ { 2 } + 3 x + 5\)

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying each term.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\)

4. The equation \(( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad\) where \(p\) is a constant has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } + 2 p - 24 > 0\)
2. Hence find the set of possible values of \(p\).

7. The equation \(2 x ^ { 2 } + 5 p x + p = 0\), where \(p\) is a constant, has no real roots. Find the set of possible values for \(p\).

10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
2. Hence find the set of all possible values of \(k\).

9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.

5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ).
2. Use the factor theorem to show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
4. Use your answer to part (c) and the sketch to deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)

8. The equation \(( k - 4 ) x ^ { 2 } - 4 x + k - 2 = 0\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
2. Find the exact range of possible values for \(k\).