Linear function

Equations

1. Solve the following equations:
   

   (a)   $8x-4=5(x+1)$

   (b)   $x^2+2x+8=(x-2)x$

   (c)   $(x+2{)}^2-(x-1{)}^2=9(x-3)$

   (d)   $(x-1)(x+2)(x+3)=x^3+4x^2+8x+16$

   Solutions:    (a)  $x=3$;     (b)  $x=-2$;     (c)  $x=10$;     (d)  $x=-\frac{22}{7}$
2. Solve the following equations:
   

   (a)   ${\displaystyle \frac{x}{3}=\frac{x+2}{4}}$

   (b)   ${\displaystyle \frac{x}{4}-4x=\frac{3x-7}{2}}$

   (c)   ${\displaystyle \frac{x+5}{3}=1+\frac{2}{3}(x+\frac{1}{2})}$

   (d)   ${\displaystyle \frac{1}{4}(x+2{)}^2-(\frac{x-1}{2}{)}^2=\frac{1}{4}+4x}$

   (e)   ${\displaystyle (\frac{x+1}{2}{)}^3=\frac{1}{8}(x^3+3x^2)-1}$

   Solutions:    (a)  $x=6$;     (b)  $x=\frac{2}{3}$;     (c)  $x=1$;     (d)  $x=\frac{1}{5}$;     (e)  $x=-3$
3. Solve the following equations:
   

   (a)   ${\displaystyle \frac{1}{2x+1}=\frac{1}{3x-2}}$

   (b)   ${\displaystyle \frac{x+1}{x-1}=\frac{3}{2}}$

   (c)   ${\displaystyle \frac{x}{x+2}+\frac{3}{x-2}=1}$

   (d)   ${\displaystyle \frac{2x}{x-2}=\frac{x+2}{x-2}}$

   Solutions:    (a)  $x=3$;     (b)  $x=5$;     (c)  $x=-10$;     (d)  No solution: $x$ doesn't exist.

Inequalities

4. Simplify the following inequalities:
   

   (a)   ${\displaystyle x+3⩽\frac{x}{3}+11}$

   (b)   ${\displaystyle \frac{x+7}{6}>\frac{1}{2}(x-\frac{1}{3})}$

   (c)   ${\displaystyle (\frac{x+2}{2}{)}^2⩽\frac{1}{4}(x+3{)}^2}$

   (d)   ${\displaystyle \frac{x+1}{2}\cdot \frac{x-2}{3}<\frac{x(x+2)}{6}}$

   Solutions:    (a)  $x⩽12$;     (b)  $x<4$;     (c)  $x⩾-\frac{5}{2}$;     (d)  $x>-\frac{2}{3}$
5. Solve the following inequalities in $\mathbb{Z}$ (write the sets of all integers which satisfy the inequalities):
   

   (a)   ${\displaystyle \frac{x+7}{3}+(x-1{)}^2⩽x^2}$

   (b)   ${\displaystyle \frac{1}{2}\cdot \frac{2-5x}{2}<x^2-(x+\frac{1}{2}{)}^2}$

   Solutions:    (a)  $x\in \{2,3,4,5,\dots \}$;     (b)  $x\in \{4,5,,6,7,\dots \}$
6. Solve the following inequalities in $\mathbb{R}$. Write the sets of solutions as intervals:
   

   (a)   ${\displaystyle \frac{2+3x}{4}⩽x^2-(x-\frac{1}{2}{)}^2}$

   (b)   ${\displaystyle \frac{x^2}{3}-\frac{x^2-2}{5}⩽\frac{2x}{3}\cdot \frac{x-3}{5}}$

   Solutions:    (a)  $x\in [3,\mathrm{\infty })$;     (b)  $x\in (-\mathrm{\infty },-1]$

Simultaneous inequalities

7. Simplify simultaneous inequalities:
   

   $3(x-3)>x-7,2(x+5)⩾6x-2$

   Solutions:    $1<x⩽3$
8. Simplify simultaneous inequalities. Write the set of all integers which satisfy simultaneously both inequalities:
   

   (a)   $5x-6<x+10,4-x⩽2(x+5)$

   (b)   ${\displaystyle \frac{1}{3}\cdot (x-\frac{1}{2})>0,x-\frac{x+3}{2}⩽\frac{x}{6}+1}$

   Solutions:    (a)  $x\in \{-2,-1,0,1,2,3\}$;     (b)  $x\in \{1,2,3,4,5,6,7\}$
9. Solve simultaneous inequalities in $\mathbb{R}$. Write the set of all solutions as an interval:
   

   ${\displaystyle \frac{1}{3}\cdot (x-\frac{1}{2})>0,x-\frac{x+3}{2}⩽\frac{x}{6}+1}$

   Solutions:    $x\in (\frac{1}{2},\frac{15}{2}]$
10. Solve simultaneous inequalities in $\mathbb{R}$:
    

    ${\displaystyle 1-\frac{x-2}{4}⩽\frac{x}{2},(x+1{)}^2<x^2+1}$

    Solutions:    No solutions, $x$ doesn't exist.

Straight line graph

11. Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following linear functions:
    

    (a)   $f(x)=2x-5$

    (b)   $f(x)=x+3$

    (c)   $f(x)=3-x$

    (d)   $f(x)=5$

    Solutions:    (a)  $m=2,c=-5$;     (b)  $m=1,c=3$;     (c)  $m=-1,c=3$;     (d)  $m=0,c=5$
12. Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following straight lines:
    

    (a)   $y=-x$

    (b)   $y=\frac{1}{2}x+3$

    (c)   $y=\frac{1}{2}+\frac{x}{3}$

    (d)   $y=\frac{3x-2}{12}$

    Solutions:    (a)  $m=-1,c=0$;     (b)  $m=\frac{1}{2},c=3$;     (c)  $m=\frac{1}{3},c=\frac{1}{2}$;     (d)  $m=\frac{1}{4},c=-\frac{1}{6}$
13. Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following straight lines:
    

    (a)   $x+y=2$

    (b)   $2x-4y+1=0$

    (c)   $\frac{x}{3}+\frac{y}{6}=1$

    Solutions:    (a)  $m=-1,c=2$;     (b)  $m=\frac{1}{2},c=\frac{1}{4}$;     (c)  $m=-2,c=6$
14. Draw the graphs of the following linear functions:
    

    (a)   $f(x)=x+2$

    (b)   $f(x)=2x-1$

    (c)   $f(x)=\frac{1}{3}x+1$

15. Draw the following straight lines:
    

    (a)   $y=x$

    (b)   $y=-2x+3$

    (c)   $x-2y=4$

    (d)   $\frac{x}{3}+\frac{y}{2}=1$

16. Straight line $\ell$ passes through points $A(1,-2)$ and $B(4,7)$. Write down the equation of this straight line.
    Solutions:    $y=3x-5$
17. Straight line $\ell$ passes through points $A(-2,-1)$ and $B(7,5)$.
    

    (a)   Write down the equation of this straight line.

    (b)   Draw this straight line in the coordinate system.

    Solutions:    (a)  $y=\frac{2}{3}x+\frac{1}{3}$
18. Straight line $\ell$ passes through points $A(\frac{1}{3},-\frac{1}{4})$ and $B(\frac{4}{3},\frac{1}{2})$.
    

    (a)   Write down the equation of this straight line.

    (b)   Find the coordinates of the $x$-axis intercept.

    (c)   Draw this straight line in the coordinate system.

    Solutions:    (a)  $y=\frac{3}{4}x-\frac{1}{2}$;     (b)  $C(\frac{2}{3},0)$
19. Straight line has the equation $4x+3y=24$.
    

    (a)   Find the coordinates of $x$-axis intercept $A$ and $y$-axis intercept $B$.

    (b)   Calculate the length of the line segment $AB$.

    (c)   Draw this straight line in the coordinate system.

    Solutions:    (a)  $A(6,0),B(0,8)$;     (b)  $AB=10$
20. Straight line ${\ell }_1$ has the equation $x-6y+3=0$.
    

    (a)   Find the gradient and the $y$-intercept of the line ${\ell }_1$.

    (b)   Write the equation of the straight line ${\ell }_2$ which is parallel to ${\ell }_1$ and passes through the origin.

    Solutions:    (a)  $m=\frac{1}{6},c=\frac{1}{2}$;     (b)  $y=\frac{1}{6}x$
21. Straight lines ${\ell }_1$ and ${\ell }_2$ have the equations    ${\ell }_1:y=\frac{1-2x}{6}$    and    ${\ell }_2:y=mx-2$.
    

    (a)   Find $m$, given that ${\ell }_1||{\ell }_2$.

    (b)   Find the area of the triangle formed by the line ${\ell }_2$ and both coordinate axes.

    Solutions:    (a)  $m=-\frac{1}{3}$;     (b)  $A=6$

Simultaneous linear equations

22. Find the point of intersection of the following two straight lines:
    

    $y=3x-2,y=\frac{1}{2}x+3$

    Solutions:    $P(2,4)$
23. Solve the simultaneous linear equations:
    

    (a)   $x+2y=10,3x+5y=27$

    (b)   $y=2x+7,4x+2y=6$

    (c)   $7x-y=5,5x+3y=-2$

    (d)   $\frac{2}{3}x+\frac{4}{3}y=\frac{2}{3},\frac{1}{2}x+\frac{3}{2}y=-1$

    Solutions:    (a)  $x=4,y=3$;     (b)  $x=-1,y=5$;     (c)  $x=\frac{1}{2},y=-\frac{3}{2}$;     (d)  $x=7,y=-3$
24. Straight lines ${\ell }_1$ and ${\ell }_2$ have the equations    ${\ell }_1:y=-2x+6$    and    ${\ell }_2:y=\frac{1}{2}x-1$.
    

    (a)   Find the point of intersection $P$.

    (b)   Find the area of the triangle formed by the lines ${\ell }_1,{\ell }_2$ and $y$-axis.

    Solutions:    (a)  $P(\frac{14}{5},\frac{2}{5})$;     (b)  $A=\frac{49}{5}=9.8$

 Index