1. Solve the following inequalities, leaving your answer in interval notation:
$\dfrac{2-5x}{6} > \dfrac{3}{5}$
$\dfrac{2x-1}{4}-\dfrac{2}{5} \geqslant 1$
$(-\infty, -\frac{8}{25})$
$[\frac{33}{10},\infty)$
2. Solve the following inequalities, leaving your answer in set notation:
$\dfrac{x}{2}+4 > \dfrac{2x-5}{6}$
$\dfrac{1}{2}(3x-1) \leqslant \dfrac{1}{4}(x+1)$
$\{x: x>-29\}$
$\{x: x \leqslant 0.6\}$
3. A student attempts to solve $\frac{1}{x} > 1$ as follows: $$\dfrac{1}{x} > 1 \Rightarrow 1 > x$$ Explain the student's mistake.
There is a problem with multiplying by $x$ - if $x$ is negative, the inequality sign needs to change direction.
4. Solve the following inequalities:
$-5 < 4x+3 \leqslant 13$
$2x+3 \leqslant 9-4x < x+14$
$-2 < x \leqslant 2.5$
$-1 < x \leqslant 1$
5. A rectangle is such that its length is 6 units greater than its width. Given that the area is at least 40, determine the range of possible values for the length of the rectangle.
At least 10
6. Find the range of values of $k$ for which $$x^2 - kx + k + 3 = 0$$ has no real roots. Give your answer using interval notation.
$(-2,6)$
7. Solve $$3x^2 + 7x + 2 > 0$$ leaving your answer in set notation.
$\{x:x < -2\} \cup \{x: x > -\frac{1}{3}\}$
8. Solve $$3x^2 + 2x - 5 \leqslant 0$$ leaving your answer in set notation.
$\{x: x\geqslant -\frac{5}{3}\} \cap \{x: x \leqslant 1\}$
9. The equation $(k+3)x^2 + 6x + k = 5$ has two distinct real solutions. Find the set of possible values of $k$.
$\{k: k > -4\} \cap \{k: k < 6\}$
10. The equation $x^2 + (k-3)x + (3-2k) = 0$ has two distinct real solutions. Find the set of possible values of $k$.
$\{k: k < -3\} \cup \{k: k > 1\}$
11. By drawing these lines on the same axes, find the set of values for which the line $y = 2x^2 + 3x - 15$ is below the line with equation $y = 8 + 2x$. Give your answer in interval notation to 2 decimal places.
$(-3.65, 3.15)$
12. Find the set of values which satisfies either $5x-1 > 3x+3$ or $2x^2 - 5x - 3 < 0$ or both.
$\{x: x>-\frac{1}{2}\}$
13. Solve $\dfrac{5}{x-3} < 2$ Give your answer using interval notation.
$(-\infty, 3) \cup (\frac{11}{2},\infty)$
14. Solve $$\dfrac{4x-3}{2-x} < 2$$
$x<\frac{7}{6}$ or $x > 2$
15. Solve $$x + 3(x^2-4x+2) > 0$$
$x < \dfrac{2}{3}$ or $x > 3$
16. Determine the range of values which satisfy both the following equations $$6 - 2(x+2) < 10$$ $$(x+1)^2 \geq 4x + 9$$
$-4 < x \leqslant -2$ or $x \geqslant 4$
17. The equation $x^2 + 4kx + 3+11k = 0$ has no real solutions. Find the possible values of $k$.
$-0.25 < k < 3$
18. The equation $x^2 + kx + 8 = k$ has no real solutions. Find the set of possible values of $k$.
$\{k: k > -8\} \cap \{k: < 4\}$
19. Solve $$\dfrac{x^2+15}{x} \geqslant 8$$
$0 < x \leqslant 3$ or $x \geqslant 5$
20. Solve $$\dfrac{3}{x+2} < x$$
$-3 < x < -2$ or $x > 1$
21. Solve $$\dfrac{x+2}{x} \geqslant x$$
$x \leqslant -1$ or $0 < x \leqslant 2$
22. Determine the range of values which satisfy both the following: $$4(2x+3) + x > 47 - 5x$$ $$(5-x)(2x+1) \leqslant 0$$
$x \geqslant 5$
23. Find the set of values which satisfy $$\dfrac{4x+1}{x-2} \geqslant 3$$
$\{x: x \leqslant -7\} \cup \{x: x > 2\}$
24. The set of values which satisfies both the following: $$5x+13 > 4(x+2)$$ $$(x-2)^2 - k(x-2)(x+3) < 0$$ is $\{x: -5 < x < -\frac{17}{4}\} \cup \{x: x > m\}$, where $k$ and $m$ are non-zero constants. Find $k$ and $m$.
