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)   \({\displaystyle\frac{x+5}{3} = 1+\frac{2}{3}\,\Big(x+\frac{1}{2}\Big)}\)

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

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

   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)   \({\displaystyle x+3 \leqslant \frac{x}{3}+11}\)

   (b)   \({\displaystyle\frac{x+7}{6} \gt \frac{1}{2}\,\Big(x-\frac{1}{3}\Big)}\)

   (c)   \({\displaystyle\Big(\frac{x+2}{2}\Big)^2 \leqslant \frac{1}{4}\,(x+3)^2}\)

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

   Solutions:    (a)  \(x\leqslant12\);     (b)  \(x\lt4\);     (c)  \(x \geqslant -\frac{5}{2}\);     (d)  \(x\gt-\frac{2}{3}\)
5. Solve the following inequalities in \(\mathbb{Z}\) (write the sets of all integers which satisfy the inequalities):
   

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

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

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

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

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

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

Simultaneous inequalities

7. Simplify simultaneous inequalities:
   

   \(3(x-3)\gt x-7,~~~~ 2(x+5)\geqslant 6x-2\)

   Solutions:    \(1\lt x\leqslant 3\)
8. Simplify simultaneous inequalities. Write the set of all integers which satisfy simultaneously both inequalities:
   

   (a)   \(5x-6\lt x+10,~~~~ 4-x\leqslant 2(x+5)\)

   (b)   \({\displaystyle\frac{1}{3}\cdot\Big(x-\frac{1}{2}\Big)\gt 0,~~~~ x-\frac{x+3}{2}\leqslant \frac{x}{6}+1 }\)

   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:
   

   \({\displaystyle\frac{1}{3}\cdot\Big(x-\frac{1}{2}\Big)\gt 0,~~~~ x-\frac{x+3}{2}\leqslant \frac{x}{6}+1 }\)

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

    \({\displaystyle 1-\frac{x-2}{4}\leqslant \frac{x}{2},~~~~ (x+1)^2\lt x^2+1 }\)

    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)   \(y=\frac{\textstyle 1}{\textstyle 2}+\frac{\textstyle x}{\textstyle 3}\)

    (d)   \(y=\frac{\textstyle 3x-2}{\textstyle 12}\)

    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)   \(\frac{\textstyle x}{\textstyle 3}+\frac{\textstyle y}{\textstyle 6}=1\)

    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)   \(\frac{\textstyle x}{\textstyle 3}+\frac{\textstyle y}{\textstyle 2}=1\)

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    \(\ell_1\!:~~ y=\frac{\textstyle1-2x}{\textstyle 6}\)    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