Combined linear and quadratic inequalities

3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.

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

1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).

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

1. \(4 x - 3 > 7 - x\)
2. \(2 x ^ { 2 } - 5 x - 12 < 0\)
3. both \(4 x - 3 > 7 - x\) and \(2 x ^ { 2 } - 5 x - 12 < 0\)

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

1. \(3 ( x - 2 ) < 8 - 2 x\)
2. \(( 2 x - 7 ) ( 1 + x ) < 0\)
3. both \(3 ( x - 2 ) < 8 - 2 x\) and \(( 2 x - 7 ) ( 1 + x ) < 0\)

1. (i) Solve the inequality

$$x ^ { 2 } + 3 x > 10 .$$ (ii) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3 \\ & x ^ { 2 } + 3 x > 10 \end{aligned}$$

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

1. \(3 x - 7 > 3 - x\)
2. \(x ^ { 2 } - 9 x \leqslant 36\)
3. both \(3 x - 7 > 3 - x\) and \(x ^ { 2 } - 9 x \leqslant 36\)

7 Solve the inequalities

1. \(- 9 \leqslant 6 x + 5 \leqslant 0\),
2. \(6 x + 5 < x ^ { 2 } + 2 x - 7\).

8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is \(\mathrm { y } = 190 - 70 \mathrm { x }\), where \(y\) cakes can be sold when the price of a cake is \(\pounds\) x.

1. Find how many cakes could be given away for free according to this model. The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for \(\pounds 1.20\) they will donate 50 cakes, but if they sell for \(\pounds 2.40\) they will donate 140 cakes. They use the linear model \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) to relate the number of cakes donated, \(y\), to the price of a cake, \(\pounds x\).
2. Find the values of the constants \(m\) and \(c\) for which this linear model fits the two data points.
3. Explain why the model is not suitable for very low prices.
4. The team would like to sell all the cakes that they donate. Find the set of possible prices that the cakes could have to achieve this.

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
5. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
6. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
7. Sketch the circle.
8. A line has equation \(y = k x + 6\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
   2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
   3. Hence find the values of \(k\) for which the line is a tangent to the circle.

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

1. \(6 x - 7 < 2 x + 3\),
2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).

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

1. \(6 x - 7 < 2 x + 3\),
2. \(\quad 2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).

1. (a) Solve the inequality

$$x ^ { 2 } + 3 x > 10$$ (b) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3 \\ & x ^ { 2 } + 3 x > 10 \end{aligned}$$

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

1. \(6 x - 11 > x + 4\),
2. \(x ^ { 2 } - 6 x - 16 < 0\),
3. both \(6 x - 11 > x + 4\) and \(x ^ { 2 } - 6 x - 16 < 0\).