2.2 Exercises

From Förberedande kurs i matematik 1

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===Exercise 2.2:8===
===Exercise 2.2:8===
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In the ''xy''-plane, fill in all the points whose coordinates <math>\,(x,y)\,</math> satisfy
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In the ''xy''-plane, shade in the section whose coordinates <math>\,(x,y)\,</math> satisfy
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Current revision

       Theory          Exercises      

Exercise 2.2:1

Solve the equations

a) \displaystyle x-2=-1 b) \displaystyle 2x+1=13
c) \displaystyle \displaystyle\frac{1}{3}x-1=x d) \displaystyle 5x+7=2x-6

Exercise 2.2:2

Solve the equations

a) \displaystyle \displaystyle\frac{5x}{6}-\displaystyle\frac{x+2}{9}=\displaystyle\frac{1}{2} b) \displaystyle \displaystyle\frac{8x+3}{7}-\displaystyle\frac{5x-7}{4}=2
c) \displaystyle (x+3)^2-(x-5)^2=6x+4 d) \displaystyle (x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2

Exercise 2.2:3

Solve the equations

a) \displaystyle \displaystyle\frac{x+3}{x-3}-\displaystyle\frac{x+5}{x-2}=0
b) \displaystyle \displaystyle\frac{4x}{4x-7}-\displaystyle\frac{1}{2x-3}=1
c) \displaystyle \left(\displaystyle\frac{1}{x-1}-\frac{1}{x+1}\right)\left(x^2+\frac{1}{2}\right)=\displaystyle\frac{6x-1}{3x-3}
d) \displaystyle \left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0

Exercise 2.2:4

a) Write the equation for the line \displaystyle \,y=2x+3\, in the form \displaystyle \,ax+by=c\,.
b) Write the equation for the line \displaystyle 3x+4y-5=0 in the form \displaystyle \,y=kx+m\,.

Exercise 2.2:5

a) Determine the equation for the straight line that goes between the points \displaystyle \,(2,3)\, and\displaystyle \,(3,0)\,.
b) Determine the equation for the straight line that has slope \displaystyle \,-3\, and goes through the point \displaystyle \,(1,-2)\,.
c) Determine the equation for the straight line that goes through the point \displaystyle \,(-1,2)\, and is parallel to the line \displaystyle \,y=3x+1\,.
d) Determine the equation for the straight line that goes through the point \displaystyle \,(2,4)\, and is perpendicular to the line \displaystyle \,y=2x+5\,.
e) Determine the slope, \displaystyle \,k\,, for the straight line that cuts the x-axis at the point \displaystyle \,(5,0)\, and y-axis at the point \displaystyle \,(0,-8)\,.

Exercise 2.2:6

Find the points of intersection between the pairs of lines in the following

a) \displaystyle y=3x+5\ and the x-axis b) \displaystyle y=-x+5\ and the y-axis
c) \displaystyle 4x+5y+6=0\ and the y-axis d) \displaystyle x+y+1=0\ and \displaystyle \ x=12
e) \displaystyle 2x+y-1=0\ and \displaystyle \ y-2x-2=0

Exercise 2.2:7

Sketch the graph of the functions

a) \displaystyle f(x)=3x-2 b) \displaystyle f(x)=2-x c) \displaystyle f(x)=2

Exercise 2.2:8

In the xy-plane, shade in the section whose coordinates \displaystyle \,(x,y)\, satisfy

a) \displaystyle y \geq x b) \displaystyle y < 3x -4 c) \displaystyle 2x+3y \leq 6

Exercise 2.2:9

Calculate the area of the triangle which

a) has corners at the points \displaystyle \,(1,4)\,, \displaystyle \,(3,3)\, and \displaystyle \,(1,0)\,.
b) is bordered by the lines \displaystyle \ x=2y\,, \displaystyle \ y=4\ and \displaystyle \ y=10-2x\,.
c) is described by the inequalities \displaystyle \ x+y \geq -2\,, \displaystyle \ 2x-y \leq 2\ and \displaystyle \ 2y-x \leq 2\,.