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PRACTISE QUESTIONS
Algebra
1) Index Laws
2) Expanding Brackets
3) Factorising
4) Negative and fractional indices
5) Surd
6) Rationalising denominators
7) Completing the square
8) Solving quadratic equations
9) Quadratic graph and discriminant
10) Simultaneous equations
11) Linear Inequality
12) Quadratic Inequalities
1) Index Laws
2) Expanding Brackets
3) Factorising
4) Negative and fractional indices
5) Surd
6) Rationalising denominators
7) Completing the square
8) Solving quadratic equations
9) Quadratic graph and discriminant
10) Simultaneous equations
11) Linear Inequality
12) Quadratic Inequalities
Algebra (Index Laws)
1. Simplify these expressions
a) $x^3 \times x^4$ b) $2x^3 \times 3x^2$ c) $\frac{k^3}{k^2}$ d) $\frac{4p^3}{2p}$ e) $\frac{3x^3}{3x^2}$ f) $(y^2)^5$ g) $10x^5 \div 2x^3$ h) $(p^3)^2 \div p^4$ i) $(2a^3)^2 \div 2a^3$ j) $(9x^2) \times 3( x^2)^3$ k) $2a^4 \times 3a^5$ l) $\frac{21a^3b^7}{7ab^4}$ m) $9x^2 \times 3(x^2)^3$ n) $ 3x^3 \times 2x^2 \times 4x^6$ o) $7a^4 \times (3a^4)^2$ p) $(4y^3)^3 \div 2y^3$ q) $2a^3 \div 3a^2 \times 6a^5$ r) $3a^4 \times 2a^5 \times a^3$ |
2) Expand and simplify if possible
a) $9(x-2)$ b) $x(x+9)$ c) $-3y(4-3y)$ d) $x(y+5)$ e) $-x(3x+5)$ f) $-5x(4x+1)$ g) $(4x+5)x$ h) $-3y(5-2y^2)$ i) $-2x(5x-4)$ j) $(3x-5)x^2$ k) $3(x+2)+(x-7)$ l) $5x-6-(3x-2)$ m) $4(c+3d^2)-3(2c+d^2)$ n) $(r^2+3t^2+9)-(2r^2+3t^2-4)$ o) $x(3x^2-2x+5)$ p) $7y^2(2-5y+3y^2)$ q) $-2y^2(5-7y+3y^2)$ r) $7(x-2)+3(x+4)-6(x-2)$ s) $5x-3(4-2x)+6$ t) $3x^2-x(3-4x)+7$ u)$4x(x+3)-2x(3x-7)$ v) $3x^2(2x+1)-5x^2(3x-4)$ |
3) Simplify these fractions:
a) $\frac{6x^4+10x^6}{2x}$ b) $\frac{3x^5-x^7}{x}$ c) $\frac{2x^4-4x^2}{4x}$ d) $\frac{8x^3+5x}{2x}$ e) $\frac{7x^7+5x^2}{5x}$ f) $\frac{9x^5-5x^3}{3x}$ |
Algebra (Expanding Brackets)
1) Expand and simplify if possible:
a) $(x+4)(x+7)$ b) $(x-3)(x+2)$ c) $(x-2)^2$ d) $(x-y)(2x+3)$ e) $(x+3y)(4x-y)$ f) $(2x-4y)(3x+y)$ g) $(2x-3)(x-4)$ h) $(3x+2y)^2$ i) $(2x+8y)(2x+3)$ j) $(x+5)(2x+3y-5)$ k) $(x-1)(3x-4y-5)$ l) $(x-4y)(2x+y+5)$ m) $(x+2y-1)(x+3)$ n) $(2x+2y+3)(x+6)$ o) $(4-y)(4y-x+3)$ p) $(4y+5)(3x-y+2)$ q) $(5y-2x+3)(x-4)$ r) $(4y-x-2)(5-y)$ |
2) Expand and simplify if possible:
a) $5(x+1)(x-4)$ b) $7(x-2)(2x+5)$ c) $3(x-3)(x-3)$ d) $x(x-y)(x+y)$ e) $x(2x+y)(3x+4)$ f) $y(x-5)(x+1)$ g) $y(3x-2y)(4x+2)$ h) $y(7-x)(2x-5)$ i) $x(2x+y)(5x-2)$ j) $x(x+2)(x+3y-4)$ k) $y(2x+y-1)(x+5)$ l) $y(3x+2y-3)(2x+1)$ m) $x(2x+3)(x+y-5)$ n) $2x(3x-1)(4x-y-3)$ o) $3x(x-2y)(2x+3y+5)$ p) $(x+3)(x+2)(x+1)$ q) $(x+2)(x-4)(x+3)$ r) $(x+3)(x-1)(x-5)$ s) $(x-5)(x-4)(x-3)$ t) $(2x+1)(x-2)(x+1)$ u) $(2x+3)(3x-1)(x+2)$ v) $(3x-2)(2x+1)(3x-2)$ w) $(x+y)(x-y)(x-1)$ x) $(2x-3y)^3$ |
Algebra (Factorising)
1) Factorise these expressions completely:
a) $4x+8$ b) $6x-24$ c) $20x+15$ d) $2x^2+4$ e) $4x^2+20$ f) $6x^2-18x$ g) $x^2-7x$ h) $2x^2+4x$ i) $3x^2-x$ j) $6x^2-2x$ k) $10y^2-5y$ l) $35x^2-28x$ m) $x^2+2x$ n) $3y^2+2y$ o) $4x^2+12x$ p) $5y^2-20y$ q) $9xy^2+12x^2y$ r) $6ab-2ab^2$ s) $5x^2-25xy$ t) $12x^2y+8xy^2$ u) $15y-20yz^2$ v) $12x^2-30$ w) $xy^2-x^2y$ x) $12y^2-4yx$ |
2) Factorise:
a) $x^2+4x$ b) $2x^2+6x$ c) $x^2+11x+24$ d) $x^2+8x+12$ e) $x^2+3x-40$ f) $x^2-8x+12$ g) $x^2+5x+6$ h) $x^2-2x-24$ i) $x^2-3x-10$ j) $x^2+x-20$ k) $2x^2+5x+2$ l) $3x^2+10x-8$ m) $5x^2-16x+3$ n) $6x^2-8x-8$ o) $2x^2+7x-15$ p) $2x^4+14x^2+24$ q) $x^2-4$ r) $x^2-49$ u) $36x^2-4$ v) $2x^2-50$ w) $6x^2-10x+4$ x) $15x^2+42x-9$ |
3) Factorise completely:
a) $x^3+2x$ b) $x^3-x^2+x$ c) $x^3-5x$ d) $x^3-9x$ e) $x^3-x^2-12x$ f) $2x^3-5x^2-3x$ g) $x^3-7x^2+6x$ h) $x^3-64x$ i) $2x^3-5x^2-3x$ j) $2x^3+13x^2+15x$ k) $x^3-4x$ l) $3x^3+27x^2+60x$ |
Algebra (Negative and fractional indices)
1) Simplify:
a) $x^3 \div x^{-2}$ b) $x^5 \div x^7$ c) $x^{\frac{3}{2}} \times x^{\frac{5}{2}}$ d) $(x^2)^\frac{3}{2}$ e) $(x^3)^\frac{5}{3}$ f) $3x^{0.5} \times 4x^{-0.5}$ g) $9x^\frac{2}{3} \div 3x^\frac{1}{6}$ h) $5x^\frac{7}{5} \div x^\frac{2}{5}$ i) $3x^4 \times 2x^{-5}$ j) $\sqrt{x} \times \sqrt[3]{x}$ k) $(\sqrt{x})^3 \times (\sqrt[3]{x})^4$ l) $\frac{(\sqrt[3]{x})^2}{\sqrt{x}}$ |
2) Evaluate, without using your calculator:
a) $25^\frac{1}{2}$ b) $81^\frac{3}{2}$ c) $27^\frac{1}{3}$ d) $4^{-2}$ e) $9^{-\frac{1}{2}}$ f) $(-5)^{-3}$ g) $(\frac{3}{4})^0$ h) $1296^\frac{3}{4}$ i) $(\frac{25}{16})^\frac{3}{2}$ j) $(\frac{27}{8})^\frac{2}{3}$ k) $(\frac{6}{5})^{-1}$ l) $(\frac{343}{512})^{-\frac{2}{3}}$ |
3) Simplify:
a) $(64x^{10})^\frac{1}{2}$ b) $\frac{5x^3-2x^2}{x^5}$ c) $(125x^{12})^\frac{1}{3}$ d) $\frac{x+4x^3}{x^3}$ e) $\frac{2x+x^2}{x^4}$ f) $(\frac{4}{9}x^4)^\frac{3}{2}$ g) $\frac{9x^2-15x^5}{3x^3}$ h) $\frac{5x+3x^2}{15x^3}$ |
Algebra (Surd) - without calculator
1) Simplify:
a) $\sqrt{28}$ b) $\sqrt{72}$ c) $\sqrt{50}$ d) $\sqrt{32}$ e) $\sqrt{90}$ f) $\frac{\sqrt{12}}{2}$ g) $\frac{\sqrt{27}}{3}$ h) $\sqrt{20}+\sqrt{80}$ i) $\sqrt{200}+\sqrt{18}-\sqrt{72}$ j) $\sqrt{175}+\sqrt{63}+2\sqrt{28}$ k) $\sqrt{28}-2\sqrt{63}+\sqrt{7}$ l) $\sqrt{80}-2\sqrt{20}+3\sqrt{45}$ m) $3\sqrt{80}-2\sqrt{20}+5\sqrt{45}$ n) $\frac{\sqrt{44}}{\sqrt{11}}$ o) $\sqrt{12}+3\sqrt{48}+\sqrt{75}$ |
2) Expand and simplify if possible:
a) $\sqrt{3}(2+\sqrt{3})$ b) $\sqrt{5}(3-\sqrt{3})$ c) $\sqrt{2}(4-\sqrt{5})$ d) $(2-\sqrt{2})(3+\sqrt{5})$ e) $(2-\sqrt{3})(3-\sqrt{7})$ f) $(4+\sqrt{5})(2+\sqrt{5})$ g) $(5-\sqrt{3})(1-\sqrt{3})$ h) $(4+\sqrt{3})(2-\sqrt{3})$ i) $(7-\sqrt{11})(2+\sqrt{11})$ 3) Simplify $\sqrt{75}-\sqrt{12}$ giving your answer in the form $a\sqrt{3}$, where $a$ is an integer. |
Algebra (Rationalising denominators) - without calculator
1) Simplify:
a) $\frac{1}{\sqrt{5}}$ b) $\frac{1}{\sqrt{11}}$ c) $\frac{1}{\sqrt{2}}$ d) $\frac{\sqrt{3}}{\sqrt{15}}$ e) $\frac{\sqrt{12}}{\sqrt{48}}$ f) $\frac{\sqrt{5}}{\sqrt{80}}$ g) $\frac{\sqrt{12}}{\sqrt{156}}$ h) $\frac{\sqrt{7}}{\sqrt{63}}$ |
2) Rationalise the denominators and simplify:
a) $\frac{1}{1+\sqrt{3}}$ b) $\frac{1}{2+\sqrt{5}}$ c) $\frac{1}{3-\sqrt{7}}$ d) $\frac{4}{3-\sqrt{5}}$ e) $\frac{1}{\sqrt{5}-\sqrt{3}}$ f) $\frac{3-\sqrt{2}}{4-\sqrt{5}}$ g) $\frac{5}{2+\sqrt{5}}$ h) $\frac{5\sqrt{2}}{\sqrt{8}-\sqrt{7}}$ i) $\frac{11}{3+\sqrt{11}}$ j) $\frac{\sqrt{3}-\sqrt{7}}{\sqrt{3}+\sqrt{7}}$ k) $\frac{\sqrt{17}-\sqrt{11}}{\sqrt{17}+\sqrt{11}}$ l) $\frac{\sqrt{41}+\sqrt{29}}{\sqrt{41}-\sqrt{29}}$ m) $\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ |
3) Rationalise the denominators and simplify:
a) $\frac{1}{(3-\sqrt{2})^2}$ b) $\frac{1}{2+\sqrt{5})^2}$ c) $\frac{4}{(3-\sqrt{2})^2}$ d) $\frac{3}{5+\sqrt{2})^2}$ e) $\frac{1}{(5+\sqrt{2})(3-\sqrt{2})}$ f) $\frac{2}{(5-\sqrt{3})(2+\sqrt{3})}$ 4) Simplify $\frac{3-2\sqrt{5}}{\sqrt{5}-1}$ giving your answer in the form $p+q\sqrt{5}$, where $p$ and $q$ are rational numbers. |
Algebra (Completing the square)
* by formula, $ax^2 \pm bx+c= a(x \pm \frac{b}{2a})^2- \frac{b^2}{4a}+c$
* by expand and compare, let $$ax^2 \pm bx+c= p(x \pm q)^2+r$$ or $$ax^2 \pm bx+c= (px \pm q)^2+r$$ where $p$, $q$ and, $r$ are contants to be found.
* by expand and compare, let $$ax^2 \pm bx+c= p(x \pm q)^2+r$$ or $$ax^2 \pm bx+c= (px \pm q)^2+r$$ where $p$, $q$ and, $r$ are contants to be found.
1) Complete the square for these expressions:
a) $x^2+4x$ b) $x^2-6x$ c) $x^2-16x$ d) $x^2+x$ e) $x^2-14x$ 2) Complete the square for these expressions: a) $2x^2+16x$ b) $3x^2-24x$ c) $5x^2+20x$ d) $2x^2-5x$ e) $8x^2-2x^2$ |
3) Write each of these expressions in the form $p(x+q)^2+r$, where $p$, $q$ and, $r$ are constants to be found:
a) $2x^2+8x+1$ b) $5x^2-15x+3$ c) $3x^2+2x-1$ d) $10-16x-4x^2$ e) $2x-8x^2+10$ 4) Given that $x^2+3x+6=(x+a)^2+b$, find the values of the constants $a$ and $b$. 5) Write $2+0.8x-0.04x^2$ in the form $A-B(x+C)^2$, where $A$, $B$, and $C$ are constants to be determined. |
Algebra (Solving quadratic equations)
We can solve a quadratic equation $ax^2+bx+c=0$ by
* factorisation
* completing the square
* by formula, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ [you may find the proof at "Proof"]
* factorisation
* completing the square
* by formula, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ [you may find the proof at "Proof"]
1) Solve the following equations using factorisation:
a) $x^2+3x+2=0$ b) $x^2+5x+4=0$ c) $x^2+7x+10=0$ d) $x^2-x-6=0$ e) $x^2-8x+15=0$ f) $x^2-9x+20=0$ g) $x^2-5x-6=0$ h) $x^2-4x-12=0$ |
2) Solve the following equations using factorisation:
a) $x^2=4x$ b) $x^2=25x$ c) $3x^2=6x$ d) $5x^2=30x$ e) $2x^2+7x+3=0$ f) $6x^2-7x-3=0$ g) $6x^2-5x-6=0$ h) $4x^2-16x+15=0$ |
3) Solve the following equations:
a) $3x^2+5x=2$ b) $(2x-3)^2=9$ c) $(x-7)^2=36$ d) $2x^2=8$ e) $3x^2=5$ f) $(x-3)^2=13$ g) $(3x-1)^2=11$ h) $5x^2-10x^2=-7+x+x^2$ i) $6x^2-7=11x$ j) $4x^2+17x=6x-2x^2$ |
4) Solve the following equations using quadratic formula. Give your answers exactly, leaving them in surd form where necessary.
a) $x^2+3x+1=0$ b) $x^2-3x-2=0$ c) $x^2+6x+6=0$ d) $x^2-5x-2=0$ e) $3x^2+10x-2=0$ f) $4x^2-4x-1=0$ g) $4x^2-7x=2$ h) $11x^2+2x-7=0$ |
5) Solving the following equations using the quadratic formula. Give your answers to three significant figures.
a) $x^2+4x+2=0$ b) $x^2-8x+1=0$ c) $x^2+11x-9=0$ d) $x^2-7x-17=0$ e) $5x^2+9x-1=0$ f) $2x^2-3x-18=0$ g) $3x^2+8=16x$ h) $2x^2+11x=5x^2-18$ |
6) For each of the equations below, choose a suitable method and find all the solutions. Where necessary, give your answers to three significant figures.
a) $x^2+8x+12=0$ b) $x^2+9x-11=0$ c) $x^2-9x-1=0$ d) $2x^2+5x+2=0$ e) $(2x+8)^2=100$ f) $6x^2+6=12x$ g) $2x^2-11=7x$ h) $x=\sqrt{8x-15}$ |
7) Solve these quadratic equations by completing the square. Leave your answers in surd form.
a) $x^2+6x+1=0$ b) $x^2+12x+3=0$ c) $x^2+4x-2=0$ d) $x^2-10x=5$ |
8) Solve these quadratic equations by completing the square. Leave your answers in surd form.
a) $2x^2+6x-3=0$ b) $5x^2+8x-2=0$ c) $4x^2-x-8=0$ d) $15-6x-2x^2=0$ |
Algebra (Quadratic graph and discriminant)
*For the quadratic function $f(x)=ax^2+bx+c$, the expression $b^2-4ac$ is called the discriminant. The value of the discriminant shows how many roots $f(x)$ has:
- if $b^2-4ac >0$ then $f(x)$ has two distinct real roots.
- if $b^2-4ac =0$ then $f(x)$ has one repeated roots.
- if $b^2-4ac <0$ then $f(x)$ has no real roots.
- When $a$ is positive, the parabola will be U-shape. When $a$ is negative, the parabola will be ∩-shape.
- When $b^2-4ac>0$, the parabola will cuts x-axis at two distinct points. When $b^2-4ac=0$, the parabola will touch the x-axis. When $b^2-4ac<0$, the parabola doesn't cut the x-axis.
- The y-intercept is equal to $c$.
- You can find the x-intercept(s) by solving $ax^2+bx+c=0$.
- The line of symmetry of parabola is equal to $x=-\frac{b}{2a}$.
- The vertex/ turning point of parabola is $(-\frac{b}{2a} , -\frac{b^2}{4a}+c)$.
1) Sketch the graphs of the following equations. For each graph, show the coordinates of the point(s) where the graph crosses the coordinate axes, and write down the coordinates of the turning point and the equation of the line of symmetry.
a) $y=x^2-6x+8$ b) $y=x^2+2x-15$ c) $y=25-x^2$ d) $y=x^2+3x+2$ e) $y=-x^2+6x+7$ f) $y=2x^2+4x+10$ g) $y=2x^2+7x-15$ h) $y=6x^2-19x+10$ i) $y=4-7x-2x^2$ j) $y=0.5x^2+0.2x+0.02$ |
2) Calculate the value of the discriminant for each of these functions:
a) $f(x)=x^2+8x+3$ b) $g(x)=2x^2-3x+4$ c) $h(x)=-x^2+7x-3$ d) $j(x)=x^2-8x+16$ e) $k(x)=2x-3x^2-4$ |
Algebra (Simultaneous equations)
* Simultaneous equations can be solved using
- by graphs
- by elimination
- by substitution
1) a) Draw the graph of $y=3x^2-x^3-1$ for $-2 \leq x \leq 3$
b) Use your graph to solve these equations. i) $3x^2-x^3-1=0$ ii) $3x^2-x^3-4=0$ iii) $3x^2-x^3-4+x=0$ 2) a) Draw the graph of $y=x^4-4x^2+2$ for $-3 \leq x \leq3$ b) Use your graph to solve these equations. i) $x^4-4x^2+1=0$ ii) $x^4-4x^2-2x+3=0$ iii) $2x^4-8x^2+x+2=0$ 3) a) Draw the graph of $y=\frac{12}{x^2}$ for $-5 \leq x \leq 5$ where $x \neq 0$ b) Use your graph to solve these equations. i) $\frac{12}{x^2}-x-2=0$ ii) $\frac{12}{x^2}+x-5=0$ iii) $12-x^3+x^2=0$ |
4) Solve the simultaneous equations graphically, drawing graphs from $-4 \leq x \leq 4$
a) $y=4-x^2, y=1+2x$ b) $y=x^2+2x-1, 1+3x-y=0$ c) $y=x^2-4x+6, y+2=2x$ d) $x^2+y=4, y=1-\frac{x}{4}$ e) $y=\frac{4}{x}, y+1=x$ f) $y=x^3+2x^2, y-1=\frac{1}{2}x$ g)$y=x^2-x-5, y=1-2x$ h) $y=2x^2-2x-4, y=6-x$ i) $y=10x^2+3x-4, y=2x-2$ j) $(x+1)^2+y=6, y=x+3$ k) $y=x^3-4x^2+5, y=3-2x$ l) $y=\frac{10}{x} +4, y=5x+2$ |
5) Solve these simultaneous equations by elimination:
a) $2x-y=6, 4x+3y=22$ b) $7x+3y=16, 2x+9y=29$ c) $5x+2y=6, 3x-10y=26$ d) $2x-y=12, 6x+2y=21$ e) $3x-2y=-6, 6x+3y=2$ f) $3x+8y=33, 6x=3+5y$ 6) Solve these simultaneous equations by substitution: a) $x+3y=11, 4x-7y=6$ b) $4x-3y=40, 2x+y=5$ c) $3x-y=7, 10x+3y=-2$ d) $2y=2x-3, 3y=x-1$ |
7) Solve the simultaneous equations:
a) $x+y=11, xy=30$ b) $2x+y=1, x^2+y^2=1$ c) $y=3x, 2y^2-xy=15$ d) $3a+b=8, 3a^2+b^2=28$ e) $2u+v=7, uv=6$ f) $3x+2y=7, x^2+y=8$ g) $2x+2y=7, x^2-4y^2=8$ h) $x+y=9, x^2-3xy+2y^2=0$ i) $5y-4x=1, x^2-y^2+5x=41$ j) $x-y=6, xy=4$ k) $2x+3y=13, x^2+y^2=78$ l) $xy=2, x^2+y^2=4$ m) $xy=1, x^2+y^2=4$ |
Algebra (Linear Inequality)
1) Find the set of values of $x$ for which:
a) $2x-3 < 5$ b) $5x+4 \geq 39$ c) $6x-3>2x+7$ d) $5x+6 \leq -12-x$ e) $15-x>4$ f) $21-2x>8+3x$ g) $1+x< 25+3x$ h) $7x-7<7-7x$ i) $5-0.5x \geq 1$ j) $5x+4 >12-2x$ |
2) Find the set of values of $x$ for which:
a) $2(x-3) \geq0$ b) $8(1-x) > x-1$ c) $3(x+7) \leq 8-x$ d) $2(x-3)-(x+12)<0$ e) $1+11(2-x)<10(x-4)$ f) $2(x-5) \geq 3(4-x)$ g) $12x-3(x-3)<45$ h) $x-2(5+2x)<11$ i) $x(x-4)\geq x^2+2$ j) $x(5-x) \geq 3+x-x^2$ k) $3x+2x(x-3) \leq 2(5+x^2)$ l) $x(2x-5) \leq \frac{4x(x+3)}{2}-9$ |
3) Use set notation to describe the set of values of $x$ for which:
a) $3(x-2) > x-4$ and $4x+12>2x+17$ b) $2x-5<x-1$ and $7(x+1) > 23-x$ c) $2x-3>2$ and $3(x+2)< 12+x$ d) $15-x< 2(11-x)$ and $5(3x-1)> 12x+19$ e) $3x+8 \leq 20$ and $2(3x-7) \geq x+6$ f) $5x+3<9$ or $5(2x+1)>27$ g) $4(3x+7) \leq 20$ or $2(3x-5) \geq \frac{7-6x}{2}$ |
Algebra (Quadratic Inequalities)
1) Find the set of values of $x$ for which:
a) $x^2-11x+24<0$ b) $12-x-x^2>0$ c) $x^2-3x-10>0$ d) $x^2+7x+12 \geq0$ e) $7+13x-2x^2>0$ f) $10+x-2x^2<0$ g) $4x^2-8x+3 \leq0$ h) $-2+7x-3x^2<0$ i) $x^2-9<0$ j) $6x^2+11x-10>0$ k) $x^2-5x>0$ l) $2x^2+3x \leq0$ |
2) Find the set of values of $x$ for which:
a) $x^2<10-3x$ b) $11<x^2+10$ c) $x(3-2x)>1$ d) $x(x+11)<3(1-x^2)$ 3) Use set notation to describe the set of values of $x$ for which: a) $x^2-7x+10<0$ and $3x+5<17$ b) $x^2-x-6>0$ and $10-2x<5$ c) $4x^2-3x-1<0$ and $4(x+2)<15-(x+7)$ d) $2x^2-x-1<0$ and $14<3x-2$ e) $x^2-x-12>0$ and $3x+17>2$ f) $x^2-2x-3<0$ and $x^2-3x+2>0$ |