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Specifying figures on the coordinate plane by equations and inequalities. Defining figures on the coordinate plane with equations and inequalities How to depict a set on the coordinate plane

Specifying figures on the coordinate plane by equations and inequalities. Defining figures on the coordinate plane with equations and inequalities How to depict a set on the coordinate plane

It is often necessary to depict on the coordinate plane the set of solutions to an inequality with two variables. A solution to an inequality with two variables is a pair of values of these variables that turns the given inequality into a true numerical inequality.

2y+ Zx< 6.

Let's draw a straight line first. To do this, we write the inequality as an equation 2y+ Zx = 6 and express y. Thus, we get: y=(6-3x)/2.

This line divides the set of all points of the coordinate plane into points above it and points below it.

Take a meme from each area checkpoint, for example A (1; 1) and B (1; 3)

The coordinates of point A satisfy the given inequality 2y + 3x< 6, т. е. 2 . 1 + 3 . 1 < 6.

Point B coordinates not satisfy this inequality 2∙3 + 3∙1< 6.

Since this inequality can change the sign on the line 2y + Zx = 6, then the inequality satisfies the set of points of the area where the point A is located. Let's shade this area.

Thus, we have depicted the set of solutions to the inequality 2y + Zx< 6.

Example

We depict the set of solutions to the inequality x 2 + 2x + y 2 - 4y + 1 > 0 on the coordinate plane.

First, we construct a graph of the equation x 2 + 2x + y 2 - 4y + 1 \u003d 0. We divide the circle equation in this equation: (x 2 + 2x + 1) + (y 2 - 4y + 4) \u003d 4, or (x + 1) 2 + (y - 2) 2 \u003d 2 2.

This is the equation of a circle centered at point 0 (-1; 2) and radius R = 2. Let's construct this circle.

Since this inequality is strict and the points lying on the circle itself do not satisfy the inequality, we construct the circle with a dotted line.

It is easy to check that the coordinates of the center O of the circle do not satisfy this inequality. The expression x 2 + 2x + y 2 - 4y + 1 changes its sign on the constructed circle. Then the inequality is satisfied by points located outside the circle. These points are shaded.

Example

Let us depict on the coordinate plane the set of solutions of the inequality

(y - x 2) (y - x - 3)< 0.

First, we construct a graph of the equation (y - x 2) (y - x - 3) \u003d 0. It is a parabola y \u003d x 2 and a straight line y \u003d x + 3. We build these lines and note that the change in the sign of the expression (y - x 2) (y - x - 3) occurs only on these lines. For the point A (0; 5), we determine the sign of this expression: (5-3) > 0 (i.e., this inequality is not satisfied). Now it is easy to mark the set of points for which this inequality is satisfied (these areas are shaded).

Algorithm for Solving Inequalities with Two Variables

1. We reduce the inequality to the form f (x; y)< 0 (f (х; у) >0; f (x; y) ≤ 0; f (x; y) ≥ 0;)

2. We write the equality f (x; y) = 0

3. Recognize the graphs recorded on the left side.

4. We build these graphs. If the inequality is strict (f (x; y)< 0 или f (х; у) >0), then - with strokes, if the inequality is not strict (f (x; y) ≤ 0 or f (x; y) ≥ 0), then - with a solid line.

5. Determine how many parts of the graphics are divided into the coordinate plane

6. Choose in one of these parts checkpoint. Determine the sign of the expression f (x; y)

7. We arrange signs in other parts of the plane, taking into account the alternation (as by the method of intervals)

8. We select the parts we need in accordance with the sign of the inequality that we are solving, and apply hatching

Let given equation with two variables F(x; y). You have already learned how to solve such equations analytically. The set of solutions of such equations can also be represented in the form of a graph.

The graph of the equation F(x; y) is the set of points of the coordinate plane xOy whose coordinates satisfy the equation.

To plot a two-variable equation, first express the y variable in terms of the x variable in the equation.

Surely you already know how to build various graphs of equations with two variables: ax + b \u003d c is a straight line, yx \u003d k is a hyperbola, (x - a) 2 + (y - b) 2 \u003d R 2 is a circle whose radius is R, and the center is at the point O(a; b).

Example 1

Plot the equation x 2 - 9y 2 = 0.

Solution.

Let us factorize the left side of the equation.

(x - 3y)(x+ 3y) = 0, i.e. y = x/3 or y = -x/3.

Answer: figure 1.

A special place is occupied by the assignment of figures on the plane by equations containing the sign of the absolute value, which we will dwell on in detail. Consider the stages of plotting equations of the form |y| = f(x) and |y| = |f(x)|.

The first equation is equivalent to the system

(f(x) ≥ 0,
(y = f(x) or y = -f(x).

That is, its graph consists of graphs of two functions: y = f(x) and y = -f(x), where f(x) ≥ 0.

To plot the graph of the second equation, graphs of two functions are plotted: y = f(x) and y = -f(x).

Example 2

Plot the equation |y| = 2 + x.

Solution.

The given equation is equivalent to the system

(x + 2 ≥ 0,
(y = x + 2 or y = -x - 2.

We build a set of points.

Answer: figure 2.

Example 3

Plot the equation |y – x| = 1.

Solution.

If y ≥ x, then y = x + 1, if y ≤ x, then y = x - 1.

Answer: figure 3.

When constructing graphs of equations containing a variable under the module sign, it is convenient and rational to use area method, based on splitting the coordinate plane into parts in which each submodule expression retains its sign.

Example 4

Plot the equation x + |x| + y + |y| = 2.

Solution.

In this example, the sign of each submodule expression depends on the coordinate quadrant.

1) In the first coordinate quarter x ≥ 0 and y ≥ 0. After expanding the module, the given equation will look like:

2x + 2y = 2, and after simplification x + y = 1.

2) In the second quarter, where x< 0, а y ≥ 0, уравнение будет иметь вид: 0 + 2y = 2 или y = 1.

3) In the third quarter x< 0, y < 0 будем иметь: x – x + y – y = 2. Перепишем этот результат в виде уравнения 0 · x + 0 · y = 2.

4) In the fourth quarter, for x ≥ 0 and y< 0 получим, что x = 1.

We will plot this equation in quarters.

Answer: figure 4.

Example 5

Draw a set of points whose coordinates satisfy the equality |x – 1| + |y – 1| = 1.

Solution.

The zeros of the submodule expressions x = 1 and y = 1 split the coordinate plane into four regions. Let's break down the modules by region. Let's put this in the form of a table.

Region	Submodule expression sign	The resulting equation after expanding the module
I	x ≥ 1 and y ≥ 1	x + y = 3
II	x< 1 и y ≥ 1	-x+y=1
III	x< 1 и y < 1	x + y = 1
IV	x ≥ 1 and y< 1	x – y = 1

Answer: figure 5.

On the coordinate plane, figures can be specified and inequalities.

Inequality graph with two variables is the set of all points of the coordinate plane whose coordinates are solutions of this inequality.

Consider algorithm for constructing a model for solving an inequality with two variables:

Write down the equation corresponding to the inequality.
Plot the equation from step 1.
Choose an arbitrary point in one of the half-planes. Check if the coordinates of the selected point satisfy the given inequality.
Draw graphically the set of all solutions of the inequality.

Consider, first of all, the inequality ax + bx + c > 0. The equation ax + bx + c = 0 defines a straight line dividing the plane into two half-planes. In each of them, the function f(x) = ax + bx + c is sign-preserving. To determine this sign, it is enough to take any point belonging to the half-plane and calculate the value of the function at this point. If the sign of the function coincides with the sign of the inequality, then this half-plane will be the solution of the inequality.

Consider examples of graphical solutions to the most common inequalities with two variables.

1) ax + bx + c ≥ 0. Figure 6.

2) |x| ≤ a, a > 0. Figure 7.

3) x 2 + y 2 ≤ a, a > 0. Figure 8.

4) y ≥ x2. Figure 9

5) xy ≤ 1. Figure 10.

If you have questions or want to practice modeling the sets of all solutions of two-variable inequalities using mathematical modeling, you can free 25-minute lesson with an online tutor after you register. For further work with the teacher, you will have the opportunity to choose the tariff plan that suits you.

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Let's call (x, y) ordered pair, and X and at are the components of this pair. At the same time, they consider that (X 1 at 1 ) = (x 2 .y 2 ), if x 1 = x 2 and at 1 = at 2 .

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Definition 9. The Cartesian product of sets A and B is called the set A B, whose elements are all pairs (x, y) such that x Ah, u B, i.e. BUT B \u003d ((x, y) / x Ah, u AT).

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Find, for example, the Cartesian product of sets A = (1,3} and B = (2,4,6).

BUT AT= {(1, 2);(1, 4);(1, 6);(3, 2);(3, 4);(3, 6)}.

The operation by which a Cartesian product is found is called Cartesian multiplication of sets.

Cartesian multiplication of sets has neither the property of commutativity nor the property of associativity, but is associated with the operations of union and subtraction of sets by distributive properties:

for any sets A, B, C equalities take place:

(BUT AT) C = (A FROM) (AT FROM),

(A\B) FROM= (BUT C)\(B FROM).

For a visual representation of the Cartesian product of numerical sets, a rectangular coordinate system is often used.

Let BUT and AT - number sets. Then the elements of the Cartesian product of these sets will be ordered pairs of numbers. Depicting each pair of numbers as a point on the coordinate plane, we get a figure that will visually represent the Cartesian product of sets BUT and AT.

Let us represent on the coordinate plane the Cartesian product of sets BUT and AT, if:

a) A = {2, 6}; B ={1,4}, b) A = (2,6}; AT= , in) A = ;B =.

In case a) these sets are finite and it is possible to enumerate the elements of the Cartesian product.

BUT B ={(2, 1); (2, 4); (6, 1); (6, 4)}. We construct the coordinate axes and on the axes OH mark the elements of the set BUT, and on the axis OU - set elements AT. Then we depict each pair of numbers in the set АВ as points on the coordinate plane (Fig. 7). The resulting figure of four points will visually represent the Cartesian product of these sets BUT and AT.

In case b) it is impossible to enumerate all elements of the Cartesian product of sets, because lots of AT- infinite, but you can imagine the process of formation of this Cartesian product: in each pair, the first component or 2 , or 6 , and the second component is a real number from the interval .

All pairs whose first component is a number 2 , and the second runs the value from 1 before 4 inclusive, are represented by segment points SD, and pairs whose first component is a number 6 , and the second is any real number from the interval , – segment points RS (Fig. 8). Thus, in case b) the Cartesian product of sets BUT and AT on the coordinate plane is depicted as a segment SD and RS.

Rice. 7 Fig. 8 Fig. 9

Case c) differs from case b) in that here not only the set AT, but also many BUT, that's why, the first component of pairs belonging to the set BUT AT, is any number from the interval . Points depicting elements of the Cartesian product of sets BUT and AT, form a square SVUL (Fig. 9). To emphasize that the elements of the Cartesian product are represented by the points of the square, it can be shaded.

test questions

Show that solving the following problems leads to the formation of a Cartesian product of sets:

a) Write down all the fractions whose numerator is a number from the set A ={3, 4} , and the denominator is a number from the set B = (5,6, 7}.

b) Write different two-digit numbers using numbers 1, 2, 3, 4.

Prove that for any sets A, B, C fair equality (BUT AT)С = (BUT FROM) (AT FROM). Illustrate its satisfiability for sets BUT= {2, 4, 6}, B=(1,3, 5), C = (0, 1).

What shape do points form on the coordinate plane if their coordinates are elements of the Cartesian product of sets BUT= (– 3, 3) and AT= R

Determine the Cartesian product of which sets BUT and AT shown in Figure 10.

Rice. ten

Exercises

112. Write down all two-digit numbers whose tens digits belong to the set BUT= {1, 3, 5} , and the digits of units - to the set B = (2,4,6).

113. Write all the fractions whose numerators are chosen from the set A=(3,5, 7}, and the denominator is from the set B={4, 6, 8}.

114. Write everything proper fractions, whose numerators are chosen from the set A =(3, 5,7), and the denominator is from the set B= (4, 6,8}.

115. Sets are given P ={1, 2, 3}, K \u003d (a,b}. Find all Cartesian products of sets R To and K R.

116. It is known that BUT AT= ((1, 2); (3, 2); (1, 4);(3, 4); (1, 6); (3, 6)). Determine what elements the sets consist of BUT and AT.

117. Write Sets (BUT AT) FROM and BUT (AT FROM) transfer steam , if BUT=(a,b}, B = {3}, C={4, 6}

118. Make sets BUT B, B BUT, if:

a )A = (a,b,s),B=(d},

b) A = { a, b}, B =  ,

in) A \u003d (t, p,k), B = A,

G) A = { x, y, z}, B = { k, n}

119. It is known that BUT B = ((2.3), (2.5), (2.6), (3.3), (3.5), (3.6)). Determine what elements the sets consist of BUT and AT.

120. Find the Cartesian Product of Sets A = {5, 9, 4} and AT= {7, 8, 6} and select from it a subset of pairs in which:

a) the first component is greater than the second; b) the first component is 5; c) the second component is 7.

121. List the elements that belong to the Cartesian product of sets A, B and FROM, if:

a) A = (2, 3}, B = (7, 8, 9}, FROM= {1, 0};

b) A = B= FROM= {2, 3};

in) BUT= {2, 3}, B = {7, 8, 9}, C = 

122. Draw on the coordinate plane the elements of the Cartesian product of sets A and B if:

a) A \u003d (x / x N,2 < X< 4}, AT= (x/x N, x< 3};

b) A \u003d (x / x R, 2 < х < 4}, В = {х/х  N, x< 3};

in) BUT= ; AT= .

123. All elements of the Cartesian product of two sets A and B are shown as points in a rectangular coordinate system. Write Sets A and AT(Fig. 11).

Rice. 13

124. Draw on the coordinate plane the elements of the Cartesian product of sets X and Y if:

a) Х=(–1.0, 1.2),Y={2, 3,4};

b) Х=(–1.0, 1.2),Y=;

in) Х = [–1;2],Y = {2, 3, 4};

G) X= , Y = ;

e) X = [–3; 2], Y = ;

and) X = ]–3;2[, Y= R;

h) X=(2),Y= R;

and) X=R, Y = {–3}.

125. The figures shown in fig. 14 are the result of the image on the coordinate plane of the Cartesian product of the sets X and Y. Specify these sets for each figure.

Rice. fourteen

126. Find out which Cartesian product of which two sets is represented on the coordinate plane as a half-plane. Consider all cases.

127. Set the Cartesian product of which two sets is depicted on the coordinate plane as a right angle, which is formed when the coordinate axes intersect.

128. On the coordinate plane, build a line parallel to the axis OH and passing through the point R(–2, 3).

129. On the coordinate plane, build a line parallel to the axis OY and passing through the point R(–2, 3). Determine the Cartesian product of which two sets is represented on the coordinate plane as this straight line.

130. On the coordinate plane, build a strip bounded by straight lines passing through points (–2, 0) and (2, 0) and parallel to the axis OY. Describe the set of points belonging to this strip.

131. Construct a rectangle on the coordinate plane, the vertices of which are points BUT(–3, 5), AT(–3, 8), FROM(7, 5), D (7, 8). Describe the set of points in this rectangle.

132. Build on the coordinate plane a set of points whose coordinates satisfy the condition:

a) X R, y= 5;

b) X= –3, at R;

in) X R, |y| = 2;

G) | x| = 3, at R;

e) X R, y≥ 4;

e) x  R, y  4;

and) X R, |y| 4;

h) | x|  4, |y| 3 ;

and) |x| ≥1, |y| ≥ 4;

to) |x| ≥ 2, y R.

133. Draw the elements of the Cartesian product of sets on the coordinate plane X and Y, if:

a) X = R, Y = {3}; b) X = R, Y = [–3; 3]; in) X = .

134. On the coordinate plane, build a figure F if

a) F= ((x, y)| x = 2, y R}

b) F= ((x, y) |x R, y = –3);

in) F= ((x, y) | x 2, u R};

G) F= ((x, y) | x TO,y≥ – 3};

e) F= ((x, y) | |x| = 2, y R};

e) F=((x,y) |x R, |y| = 3).

135. Construct a rectangle with vertices at points (–3, 4), (–3, –3), (1, –3), (1, 4). Specify the characteristic property of the points belonging to this rectangle.

136. On the coordinate plane, construct straight lines parallel to the OX axis and passing through the points (2, 3) and (2, -1). Set the Cartesian product of which two sets is displayed on the coordinate plane as a strip enclosed between the constructed lines.

137. On the coordinate plane, construct lines parallel to the OY axis and passing through the points (2, 3) and (–2, 3). Set the Cartesian product of which two sets is displayed on the coordinate plane as a strip enclosed between the constructed lines.

138. Draw a set in a rectangular coordinate system X Y, if:

a) X = R; Y ={ y at R, |at| < 3},

b) X= {x/ x R, |X| > 2}; Y= (y/y R, |at| > 4}.

For this chapter, the student should be able to:

Define sets in different ways;

Establish relationships between sets and depict them using Euler-Venn diagrams;

Prove the equality of two sets;

Perform operations on sets and illustrate them geometrically using Euler-Venn diagrams;

Split the set into classes using one or more properties; evaluate the correctness of the classification performed.