Module 1:Linear Inequalities in Two Variables

Linear Inequalities in Two Variables

Objectives:

1. Differentiates linear inequalities in two variables from linear equations in two variables.

 

2. Illustrates and graphs linear inequalities in two variables.

 

3. Solves problems involving linear inequalities in two variables.

 

Budget of Work for Week 1:

• Translates statements from real life situations to mathematical equations or inequalities

• Differentiates linear inequalities in two variables from linear equations in two variables.

• Illustrates and graphs linear inequalities in two variables on a rectangular coordinate plane.

• Designs model to solve real life problems involving linear inequalities in two variables.

Lesson 1.1 Mathematical Expression and Equations in Two Variables

• A Mathematical expression is a combination of symbols that can designate numbers(constants), variables, operations, symbols of grouping and other punctuation. It is also the left or the right member of any mathematical statement.

• Examples:   4x + 5; 2a - 4; 15r; 6x = 4y - 20

Mathematical Expression and Equations in Two Variables

Mathematical Equation is a mathematical statement that asserts the equality of two expressions, which are connected by the equals sign "=".

Mathematical Expression and Equations in Two Variables

Linear Inequality in two variables is a mathematical statement that makes use of inequality

symbols such as  < ( lesser than), 

 > (greater than),  

≤ ( lesser than or equal to),  

≥ (greater than or equal to).

Multiple Choice

Translate: The difference between the weight of John (j) and Alex (a) is at least 28.

• j + a ≥ 28

• j - a ≥ 28

• j + a ≤ 28

• j + a ≤ 28

Multiple Choice

CHOOSE THE APPROPRIATE TERM TO DESCRIBE THE MATH SENTENCE.

6g < 23

• EXPRESSION

• EQUATION

• INEQUALITY

Multiple Choice

CHOOSE THE APPROPRIATE TERM TO DESCRIBE THE MATH SENTENCE.

19 + x = 4

• EXPRESSION

• EQUATION

• INEQUALITY

Multiple Choice

CHOOSE THE APPROPRIATE TERM TO DESCRIBE THE MATH SENTENCE.

5 > 3

• EXPRESSION

• EQUATION

• INEQUALITY

Multiple Choice

CHOOSE THE APPROPRIATE TERM TO DESCRIBE THE MATH SENTENCE.

5x + 3y

• EXPRESSION

• EQUATION

• INEQUALITY

Multiple Choice

AN ALGEBRAIC RELATION SHOWING THAT A QUANTITY IS GREATER THAN OR LESS THAN ANOTHER QUANTITY.

• EXPRESSION

• EQUATION

• INEQUALITY

Multiple Choice

Which of these can be written as an equation?

• Product of 7 and n is less than 11
• 5 times x minus 9
• Half of 8 plus b is 35
• 2 times w plus 14

Multiple Choice

Translate:

The number of Orange (o) more than twice the number of Mangoes (m) is greater than 30.

• o + 2m > 30

• o - 2m > 30

• o + 2m < 30

• o - 2m < 30

Multiple Choice

5y - 7 is an example of an ____________

• expression

• equation

Multiple Choice

Jean bought 2-blouse (b) and three shirts (s) are paid not more than Php1, 200. Translate it to mathematical statement.

• 2b - 3s ≤ 1200

• 2b + 3s ≥ 1200

• 2b -3s ≥ 1200

• 2b + 3s ≤ 1200

Lesson 1.2: Equations and Inequalities in Two Variables

• Linear Inequality in Two Variables is an inequality that can be written in one of the following forms: Ax + By < C; Ax + By > C ; Ax + By ≤ C and Ax + By ≥ C where A, B, and C are areal numbers and A and B are not both equal to zero.

• Examples: x - 5y < 4; x + 2y > 7; a + b ≤ 9; 10q - 4r ≥ 25

• Linear Equation in Two Variables is an equation of the form ax +by + c =0 where a, b & c are real numbers and a & b are not both equal to zero. Always remember that linear equations in two variables can be written in any form- standard form ex. 2x + y = 16;

  slope-intercept form ex. y = 3x - 28 or general form ex. x - 5y + 3 = 10

Lesson 1.2: Equations and Inequalities in Two Variables

• As shown in the table above only situation number 2 represents Linear Equation and the other situation represent linear inequalities:

• Situation number 1 uses an inequality symbol and it is written in the form Ax +By < C.

• Situation number 2 also illustrates linear inequality x + 5y > 200 , where x is the number of 1 -peso coins and y is

  the number for 5-peso coins.

• Lastly, situation number 3 ; 2x + y ≤ 1000 where x is the cost of each of the blouses and y is the cost of a pair of sandals

Multiple Choice

Which of the following inequalities illustrate “the sum of 𝑥 and 𝑦 is at most 15”?

• 𝑥 + 𝑦 = 15

• 𝑥 + 𝑦 > 15

• 𝑥 + 𝑦 ≤ 15

• 𝑥 + 𝑦 ≥ 15

Multiple Choice

A number of Apples (a) more than thrice the number of berries (b) is greater than 30.

• a +3b > 30

• a +3b > 30

• a + 3b≤ 30

• a +3b≥ 30

Multiple Choice

Which of the following is a linear inequality in two variables?

• 𝑥 + 𝑦 = 2

• 𝑥 + 5 > 8

• 3𝑚 + 5𝑛 < 10

• 3𝑞 – 2𝑞 ≠ 6

Multiple Choice

Six times the number of male teachers (m) is the number of female teachers (f).

• 6m > f

• 6m < f

• 6m = f

Multiple Choice

The difference between Elias height and Ronald height is not more than 1.5 ft. Which mathematical statement represents the difference

in the height of Elias and Ronald?

• x - y ≥ 1.5

• x - y ≤ 1.5

• x - y < 1.5

• x - y > 1.5

Lesson 1.3: Graphs of Linear Inequalities in Two Variables

• The graph of a linear inequality in two variables is the set of all points in the rectangular

  coordinate system in which ordered pairs satisfy the inequality.

• When a line is graphed in the

  coordinate plane, it separates the plane in two regions called half-planes (upper and lower).

Lesson 1.3: Graphs of Linear Inequalities in Two Variables

• Graph the resulting equation with solid line if the original inequality contains ≤  or ≥  symbol.

• The solid line indicates that all points on the line are part of the solution of the inequality.

Lesson 1.3: Graphs of Linear Inequalities in Two Variables

• If the inequality contains < or > symbol, use a dash or a broken line.

• The dash or broken line indicates that the coordinate of all points on the line are not part of the solution of the inequality.

Lesson 1.3: Graphs of Linear Inequalities in Two Variables

• Finally, pick one point not on the line

  ((0,0) or the origin is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

• 
  

The points found on the shaded area are the answers to the inequalities.

Multiple Choice

For this symbol,
 $\le$ 
is it a solid line or a dashed line?

Multiple Choice

For this symbol,
 $\ge$ 
do you shade above or below the line?

Multiple Choice

For this symbol,
 $\le$ 
do you shade above or below the line?

Multiple Choice

Which of the following is a solution of the inequality
 $y>\frac{1}{3}x+7$  

• (-3, 2)

• (-6, 5)

• (-9, 6)

• (0, 7)

Multiple Choice

Consider the function y<2x+3. Which is true?

• The line would be solid with shading above.
• The line would be dashed with shading above.
• The line would be solid with shading below.
• None of these.

Multiple Choice

Which point below is NOT part of the solution set?

• (0, 0)
• (-10, -10)
• (-1, 5)
• (5, 20)

Multiple Choice

What type of line is used with the following symbols, < and >?

• dashed or dotted

• solid

Multiple Choice

What type of line is used with the following symbols, < and >?

• dashed or dotted

• solid

Multiple Choice

Is the point (3, 1) a solution for the following inequality? Choose the best answer that has the best explanation.

• Yes because it is located in the shaded region.

• No because it is located on the line and the line is dotted.

• Yes because it is located on the line.

• No because it is in the shaded region.

Multiple Choice

How can I tell if the graph of an inequality will have a solid line?

• It has just an equal sign.
• The symbol is greater than.
• The symbol has "or equal to."
• You just guess.