Equations and inequalities

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Równania i nierówności

Learning objectives

You will communicate in English, develop mathematical and basic scientific and technical and IT competences, develop learning skills.
You will solve equations with the equivalent equation method.
You will solve inequalities using the equivalent inequality method.

Learning effect

You will learn to solve equations with the equivalent equation method.
You will learn to solve inequalities using the equivalent inequality method.

Think what you remember about equations and inequalities.

The aim of the lesson is to review and consolidate knowledge of equations and inequalities and methods of solving them.

Task 1

Analyze the interactive illustration which illustrates the method of solving the equationequationequation. Your task is to note down the stages of solving the equationequationequation. Formulate the conclusions.

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Conclusion:

Stages of solving the equation.

The unknown are transferred to the left side of the equationequationequation and the numbers to the right.
By moving the expression to the other side of the equationequationequation, we change its sign to the opposite one.
We carry out the reduction of similar terms.
If there is a number with the unknown, we divide both sides of the equationequationequation by this number.

Think - What additional steps should be performed if the numeric coefficients are written in the form of fractions or the expression contains brackets?

Conclusion:

If in the equationequationequation appear numerical coefficients in the form of fractions, then we can multiply both sides of the equationequationequation by the common denominator of these fractions.
We get rid of the brackets by performing the indicated operations.

Work independently and check your skills by solving the following problems.

Task 2

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Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Calculate the equations.

a) $\frac{x + 3}{2} - \frac{x - 4}{3} = 0$

b) $\frac{2 + x}{3} + \frac{3 - x}{6} = 1$

c) $\frac{x - 4}{3} - \frac{2 x - 1}{5} = x + \frac{11 - 2 x}{15}$

d) $\frac{3 (x - 11)}{4} = \frac{3 (x + 1)}{5} - \frac{2 (2 x - 5)}{11}$

Task 3

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Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

For which value of the number a, the solution of the given equationequationequation is the number $0, 5$ ?

$(3 x + 1) (x + a) - 3 (x - 1) (x - a) = 4$

Task 4

Analyze the interactive illustration that illustrates how to solve inequalities. Your task is to note down the stages of the solution of inequalityinequalityinequality. Formulate the conclusions.

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Conclusion:

Stages of solving inequality.

We transfer the unknowns to the left of the inequalityinequalityinequality and the numbers to the right.
By moving the expression to the other side of the inequalityinequalityinequality, we reverse its symbol to the opposite one.
We carry out the reduction of similar terms.
If there is a number with the unknown, we divide both sides of the inequalityinequalityinequality by that number.
Dividing the sides of inequalityinequalityinequality by a negative number requires changing the direction of the inequalityinequalityinequality to the opposite one.

Work independently and check your skills by solving the following problems.

Task 5

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Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Solve the inequality and mark the set of the solutions on the numerical axis.

a) $2 (x - 1) - 3 (x - 2) < 6$

b) $x - \frac{3 (x + 2)}{4} > 1 + x$

c) $\frac{x - 5}{2} - \frac{3 - x}{6} \leq \frac{x + 4}{3}$

d) $\frac{4 - x}{5} - \frac{x + 2}{3} \geq \frac{2 - 8 x}{15}$

Task 6

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Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

For what value of the number a, a set of solutions of inequalityinequalityinequality $2 (x - 5) < a x - 16$ is the set $(1, \infty)$ ?

Task 7

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Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

An extra task:
Solve the inequality $1 - \frac{x - 3}{3} \leq {(x - 1)}^{2} - x^{2}$ .

a) Indicate integers greater than -3 belonging to the set of inequalityinequalityinequality solutions.

b) Write three numbers that are not rational that belong to the set of inequalityinequalityinequality solutions.

c) Is the $\frac{1}{7}$ number a part of the set of inequalityinequalityinequality solutions?

Do the revision tasks.

Remember:

Stages of solving the equation (inequality).

The unknowns are transferred to the left side of the equationequationequation (inequalityinequalityinequality), and the numbers to the right.
By moving the expression to the other side of the equationequationequation (inequalityinequalityinequality), we reverse its symbol to the opposite one.
We carry out the reduction of similar terms.
If there is a number with the unknown, we divide both sides of the equationequationequation (inequalityinequalityinequality) by that number.
Dividing the sides of inequalityinequalityinequality by a negative number requires changing the direction of this inequalityinequalityinequality to the opposite one.

Exercises

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Exercise 1

The solution of the equation 40x + 8 = 0 is the number.

–5
–0,2
0,2
5

Exercise 2

In the triangle, the measure of the largest angle is three times the measure of the smallest angle. The measure of the third of the angles is 40 greater than the size of the smallest angle. Find the measure of the largest angle.

Exercise 3

Describe the stages of the inequality solution in English.

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Exercise 4

Match Polish terms with their English equivalents.

equation, solution of the equation, set of inequality solutions, right side of the equation, inequality

zbiór rozwiązań nierówności
nierówność
rozwiązanie równania
prawa strona równania
równanie

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Match Polish terms with their English equivalents.

inequality
rozwiązanie równania
równanie
left side of the equation
solution of the equation
lewa strona równania
nierówność
equation
set of inequality solutions
zbiór rozwiązań nierówności

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

equation

równanie

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Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: equation

inequality

nierówność

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Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: inequality

left side of the equation

lewa strona równania

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: left side of the equation

right side of the equation

prawa strona równania

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Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: right side of the equation

set of inequality solutions

zbiór rozwiązań nierówności

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Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: set of inequality solutions

solution of the equation

rozwiązanie równania

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Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: solution of the equation

Keywords

equationequationequation

inequalityinequalityinequality

right side of the equationright side of the equationright side of the equation

set of inequality solutionsset of inequality solutionsset of inequality solutions

solution of the equationsolution of the equationsolution of the equation

Scenariusz

Exercises

Glossary

Keywords