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Mathematics 8.

Inhaltsverzeichnis
Algebra
Algebraic expressions (reminder)
How to solve equations and inequalities? (revision)
Multiplication of algebraic expressions formed by multiple terms
The square of a sum or a difference (supplementary material)
The product of a sum and a difference (supplementary material)
Factorisation
Algebraic fractions (complementary topics)
Solving equations by factorisation
Mixed exercises
Word problems
Sets, combinatorics
Geometry I.
Three-dimensional geometry
Statistics, probability
Geometry II.
Functions, sequences
How to solve equations and inequalities? (revision)
Equations express the equality between two algebraic expressions. When writing an equation, you should always define the universal set, which is a set of numbers among which the solution of the equation can be found. If the universal set is not determined,  then the largest set within which the equation can be solved is the set of the exercise.
Example 1
Solve the following equations in the set of rational numbers.
a) \latex{ 8-4a+10 = 3a+2+a }                  b) \latex{ \frac{y}{2}+\frac{4}{5}=1.2 }
Solution
a)  \latex{ 8-4a+10 = 3a+2+a }   combine like terms 
\latex{ 18-4a = 4a+2 \;\;\;\;\;\;\;\;\;\;\;\;\;/ -4a }
\latex{ 18-8a = 2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;/ -18 }
\latex{ -8a = -16 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;/ \div (-8) }
\latex{ a = 2 }                        rational number
Check: 
left side: \latex{ 8 - 4 \times 2 + 10 = 8 - 8 + 10 = 10 };
right side: \latex{ 3 \times 2 + 2 + 2 = 6 + 2 + 2 = 10 }.
Answer: The solution is \latex{ a = 2 }.

b)  \latex{ \frac{y}{2}+\frac{4}{5} =1.2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; / -\frac{4}{5} }

  \latex{ \frac{y}{2} =1.2-0.8 }

  \latex{ \frac{y}{2} =0.4 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; / \times 2 }

  \latex{ \frac{y}{2} =0.8 }                        rational number

 

Check:  \latex{ \frac{0.8}{2}+\frac{4}{5}=0.4+0.8=1.2 }.
Answer: The solution is \latex{ y=0.8 }.

If you add the same number or the same number of variables to both sides of an equation, the sides remain equal. If you subtract the same number or the same number of variables from both sides of an equation, the sides remain equal.

If you multiply both sides of an equation by the same number other than \latex{ 0 }, the sides remain equal. If you divide both sides of an equation by the same number other than \latex{ 0 }, the sides remain equal.

 

\latex{\frac{4}{5}=0.8}

The equations were solved using the balance method, which means that the same operations were performed on both sides of the equations.
Checking the solution:
no

Determine the value

of the variable

Is the solution an

element of the universal set?

Check
Answer
The equation has no solution
yes
Example 2
Fill in each coloured square with the sum of the two values located diagonally in the row above. What number does the \latex{ x } denote?
\latex{ 59 }
\latex{ 4 }
\latex{ 5 }
\latex{ x }
\latex{ 7 }
\latex{ 9 }
Solution
\latex{\fcolorbox{black}{#fabfba}{\textcolor{#fabfba}{O}}} \latex{ = 5+x }
\latex{\fcolorbox{black}{#c7ddf3}{\textcolor{#c7ddf3}{O}}} \latex{ = x+7 }
\latex{\fcolorbox{black}{#fdd1b0}{\textcolor{#fdd1b0}{O}}} \latex{ = 9+ \fcolorbox{black}{#fabfba}{\textcolor{#fabfba}{O}} = 9 + (5+x) = 9+5+x = 14+x}
\latex{\fcolorbox{black}{#c0e2ca}{\textcolor{#c0e2ca}{O}}} \latex{ = \fcolorbox{black}{#fabfba}{\textcolor{#fabfba}{O}} + \fcolorbox{black}{#c7ddf3}{\textcolor{#c7ddf3}{O}} = (5+x) + (x+7) = 5+x + x+7 = 2x + 12 }
\latex{\fcolorbox{black}{#dacae3}{\textcolor{#dacae3}{O}}} \latex{ = \fcolorbox{black}{#fdd1b0}{\textcolor{#fdd1b0}{O}} + \fcolorbox{black}{#c0e2ca}{\textcolor{#c0e2ca}{O}} = (14 + x) + (2x+12) = 14+x+2x+12 = 3x + 26. }
The value of the purple square is \latex{ 59 }; thus,                                            Check:
\latex{ 3x + 26 = 59\;\;\;\;\;\;\;\;\;\;\;\;\; / -26 }
          \latex{ 3x = 33 \;\;\;\;\;\;\;\;\;\;\;\;/ \div 3 }
            \latex{ x=11 }
Answer: The value of \latex{ x } is \latex{ 11 }.
\latex{ 59 }
\latex{ 4 }
\latex{ 5 }
\latex{ x }
\latex{ 7 }
\latex{ 25 }
\latex{ 9 }
\latex{ 16 }
\latex{ 18 }
\latex{ 34 }
Example 3
Three numbers were written on a piece of paper. Aaron subtracted the second number from the first, and then subtracted the third number from the difference. Ben subtracted the third number from the second, and then subtracted the difference from the first number. The number Ben got was \latex{ 12 } greater than the number Aaron got. What was the third number on the paper?
Ben's result is \latex{ 12 } greater than Aaron's.
\latex{ = }
If you add \latex{ 12 } to Aaron's number, you get Ben's.
\latex{ = }
Solution
Denote the numbers by letters: number \latex{ 1 } by a; number \latex{ 2 } by \latex{ b }; number \latex{ 3 } by \latex{ c }. Aaron performed the operation \latex{ (a-b) -c }; while Ben performed the operation \latex{ a - (b-c) }.  
Ben's result is \latex{ 12 } greater than Aaron's; therefore, if you add \latex{ 12 } to Aaron's result, the two numbers will be equal.

The difference between Ben's result and Aaron's result is \latex{ 12 }.

\latex{ (a-b) = a-b }
\latex{ -(b-c) = -b+c }
\latex{ (a - b) - c + 12 = a - (b - c) }
\latex{ a - b - c + 12 = a - b + c }
\latex{ - b - c + 12 = - b + c }
  \latex{ - c + 12 =c }
        \latex{ 12 = 2c }
        \latex{ 6=c }

Remove the parentheses.

\latex{ / - a }

\latex{ / + b }

\latex{ / + c }

\latex{ / \div 2 }

You can check your answer to a word problem by substituting the solution into the text of the exercise.
Check:       Aaron: \latex{ (a - b) - 6) = a - b - 6; }
  Ben: \latex{ (a - (b - 6) = a - b + 6. }
Ben's number is indeed \latex{ 12 } greater than Aaron's, since
\latex{ a - b + 6 - (a - b - 6) = a - b + 6 - a + b + 6 = 12. }
Answer: The third number is \latex{ 6 }.
Solve the exercise with different numbers as well.
Before using the balance method, it is recommended to remove the parentheses and combine like terms on both sides of the equation.
Example 4
The sum of one-half, one-third, and one-fourth of a number equals one-tenth. What is the number?
Solution
Denote the number by \latex{ x }. According to the text, the equation is the following: 
 
\latex{ \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = \frac{1}{10} }.
 
Combine the like terms by expanding the fractions to the common denominator.
 
\latex{ \frac{6x+4x+3x}{12} = \frac{1}{10} }
 
\latex{ \frac{13x}{12} = \frac{1}{10} }
 
\latex{ 13x = \frac{\overset{6}{\cancel{12}}}{\underset{5}{\cancel{10}} } }
 
\latex{ x = \frac{6}{5 \times 13} }
 
\latex{ x = \frac{6}{65} }.
 
Check:
One-half of \latex{\frac{6}{65}}  is   \latex{ \frac{1}{\underset{1}{\cancel{2}} } \times \frac{\overset{3}{\cancel{6}}}{65} = \frac{3}{65} },
its one-third is   \latex{ \frac{1}{\underset{1}{\cancel{3}} } \times \frac{\overset{2}{\cancel{6}}}{65} = \frac{3}{65} },
its one-fourth is   \latex{ \frac{1}{\underset{2}{\cancel{4}} } \times \frac{\overset{3}{\cancel{6}}}{65} = \frac{3}{130} }.
 
\latex{ \frac{3}{65} + \frac{2}{65} + \frac{3}{130} = \frac{6+4+3}{130} = \frac{13}{130} = \frac{1}{10}}

Answer: The number is \latex{\frac{6}{65}}.

One-half of x is:

\latex{ \frac{1} {2}\times x=\frac{x}{2} }

 

one-third of x is:

\latex{ \frac{1} {3}\times x=\frac{x}{3} }

 

one fourth of x is:

\latex{ \frac{1} {4}\times x=\frac{x}{4} }

 

Example 5
Solve the following equations in the set of whole numbers.
 
a) \latex{ \frac{x}{3} - \frac{3}{4} = \frac{1}{4} + \frac{x}{2}}                 b) \latex{ \frac{x}{2} - \frac{x-3}{2} = 3}
Solution
a)
    \latex{ \frac{x}{3} - \frac{3}{4} = \frac{1}{4} + \frac{x}{2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; /- \frac{x}{3}}

         \latex{ - \frac{3}{4} = \frac{1}{4}+ \frac{x}{2} - \frac{x}{3}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; /- \frac{1}{4}} 

\latex{ - \frac{3}{4} - \frac{1}{4} = \frac{x}{2} - \frac{x}{3} }

       \latex{ - \frac4{}{4} = \frac{3x-2x}{6} }

          \latex{ - 1 = \frac{x}{6}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; /\times 6 }

          \latex{ - 6 = x }                                         whole number
 
Check:           left side: \latex{ \frac{-6}{3}-\frac{3}{4}=|-2-\frac{3}{4}=-\frac{11}{4} }
 
                        rigth side:  \latex{ \frac{1}{4}+\frac{-6}{2}=\frac{1}{4}-3=\frac{1-12}{4}=-\frac{11}{4} }
 
Answer:                                \latex{ x=-6 }

b) 
             \latex{ \frac{x}{2}-\frac{x-3}{2} = 3 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; /\times 2 }
 
\latex{ 2 \times \bigg( \frac{x}{2}-\frac{x-3}{2} \bigg) = 2 \times 3 }
 
\latex{ \cancel{2}\times \frac{x}{\cancel{2}} - \cancel{2} \times \frac{x-3}{\cancel{2}} = 6 }
 
             \latex{ x-(x-3)=6 }
                \latex{ x-x+3=6 }
                               \latex{ 3 = 6}
 
\latex{ 3 \neq 6 }; thus, this equation has no solution.

You can check the solution of an equation by replacing the variables with the value of the solution on both sides. If the sides are equal after substitution, the solution is correct.

 

In the numerator, the algebraic expression formed by a group of terms should be put into parentheses after factorisation. 

The balance method can also be used to rearrange formulas used in physics.  
For example, in uniform linear motion, speed is the quotient of distance and time: \latex{ v=\frac{s}{t} }.
If you multiply both sides of the equation by \latex{ t }, you get
\latex{ t \times v = s }, which means that distance is the product of speed and time.
 
Then, if you divide both sides of the equation by \latex{ v }, you get
\latex{ t = \frac{s}{v} } , which means that time is the quotient of distance and speed. 
 
Inequalities are used to describe relations between expressions that are not equal. The following symbols are used to express inequalities:
\latex{ (\lt; \leq; \gt; \geq; \neq ) }. 
The universal set should be defined in the case of inequalities as well.

\latex{ v=\frac{s}{t} }          \latex{ / \times t}

\latex{ v \times t = s }    \latex{ / \div v }

\latex{ t=\frac{s}{v} }

Example 6
 
Solve the following inequalities in the set of whole numbers.
a) \latex{ 3x - 2 \lt 16 }, in the set of positive integers; 
b) \latex{ 7 - 2x \geq 11 }, in the set of rational numbers.
Solution
a)                            \latex{ 3x- 2 \lt 16 }                        \latex{ /+2 }
                                       \latex{ 3x \lt 18}                      \latex{ /\div3 }
                                         \latex{ x \lt 6}
 
Solving the inequality: \latex{ x } is an element of the set of positive integers and \latex{ x \lt 6 }, so the solution set is  \latex{ \left\{1, 2, 3, 4, 5\right\} } . 
The solution of the inequality is represented on a number line:
\latex{7}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}

b)                                \latex{ 7- 2x \leq 11 }                     \latex{ /-7 }

                                  \latex{ - 2x \leq 4}                     \latex{ /\div(-2) }

                                        \latex{ x \geq -2}

 

Solving the inequality: x is a rational number and \latex{ x=\gt 2 }.The solutions to the inequality are rational numbers coloured in red on the number line.

\latex{4}
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}

\latex{ -2 \lt 3 }     \latex{ / \times (-1) }

\latex{ 2 \gt -3 }

 

Multiplying both sides of an inequality by a negative number reverses the inequality symbol.

 

\latex{ -x \leq 2 }     \latex{ / \times (-1) }

\latex{ x \geq -2 }

 

The inequality has an infinite number of solutions. 

The inequality can only be checked by substitution if there is a finite number of solutions.
The balance method can also be used to solve inequalities. Remember that multiplying or dividing both sides by a negative number reverses the inequality.
Exercises
{{exercise_number}}. Solve the following equations in the set of whole numbers.
  1. \latex{ 5x - 7 = 18 }
  1. \latex{ 3 = 7 - 2x }
  1. \latex{4 - 3x = x + 16}
  1. \latex{ 11 - 4x = 11 + 3x }
  1. \latex{ 8x + 41 = 11x + 16 }
  1. \latex{ x + 1 + 2x+ 2 + 3x+ 3 = 4x + 4 }
  1. \latex{ 3x + 7 - 4x + 9 = 5x + 13 - 8x - 1 }
  1. \latex{ \frac{x}{5} +2 = 12-x }
  1. \latex{ 2x + \frac{5}{3} = \frac{x}{6} }
  1. \latex{ \frac{x}{4} + \frac{3}{5} = \frac{5x}{20} +1}
{{exercise_number}}. Solve the following equations in the set of rational numbers.
  1. \latex{ \frac{x}{4} - 1 = \frac{37}{5}-5x}
  1. \latex{ \frac {3}{2}x + 3 = 5x+1-x}
  1. \latex{ 2(x + 2) - 1 = 4}
  1. \latex{ 4x + 2(x + 3) = 6 }
  1. \latex{ 3(x - 3) + 4(x + 1) = 2(x + 9) }
  1. \latex{ 5 - 2(x + 4) = x + 6 }
  1. \latex{ 5(x + 4) - 3(x - 2) = 20 }
  1. \latex{ 2- \frac{x+2}{6} =1 }
  1. \latex{ \frac{x+1}{3} + \frac{x-1}{2} =x-1 }
  1. \latex{ \frac{x}{3} + \frac{x-2}{4} - \frac{x+2}{5} = \frac{x}{6} +3 }
{{exercise_number}}. Match the equations with their solutions.
\latex{ \frac{x}{2} - \frac{x-2}{12} = \frac{x+7}{3} }
\latex{ x + 7 = 5x - 3 }
\latex{ -2 }
\latex{ (x - 1)x -1 = x + 2x + 5 }
\latex{ 3(x + 2) - 5(x + 3) = 9(x – 1) }
\latex{ 26 }
\latex{ 2.5 }
\latex{ 0 }
{{exercise_number}}. Solve the following equations in the set of natural numbers.
  1. \latex{ 3x - 1 \lt 8 }
  1. \latex{ 2x - 5 \leq x + 4 }
  1. \latex{ 14 - 2x \gt x - 11 }
  1. \latex{ 5 - 3x \geq 5 }
  1. \latex{ 6x + 6 - 4x \lt 8x - 18 }
  1. \latex{ 3(x - 1) + 4x \gt 3(2x - 5) }
{{exercise_number}}. Solve the following equations in the set of rational numbers. Represent each solution on a number line.
  1. \latex{ 6x-7 \geq 5}
  1. \latex{ 3 + 3x \lt x + 17 }
  1. \latex{ 18 - 6x \geq 5x + 7 }
  1. \latex{ 2(3 - x) + 5(x - 1) \leq x + 8 }
  1. \latex{ \frac{x}{3} -4 \lt \frac{x}{6} }
  1. \latex{ \frac{2-3x}{5} + \frac{x-1}{3} +1 \gt 0 }
{{exercise_number}}. Which physical quantities are expressed by the following equations? Determine them by rearranging the formulas.
  1. \latex{ p=\frac{m}{v} }
  1. \latex{ p=\frac{F_{ny}}{A} }
  1. \latex{ W= F \times s}
  1. \latex{ Q = c \times m \times t}
Quiz
Solve the equation  \latex{ x+1+2x+2+3x+3+...+99x+99 = 100x+100 }.

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