LINEAR INEQUALITIES IN TWO VARIABLES

LINEAR INEQUALITIES IN TWO VARIABLES

REVIEW: Linear Inequality in One variable

Multiple Choice

When you graph an inequality, you used a CLOSED DOT when you use which symbols?

• $\le,\ge$

• $<,>$

Multiple Choice

When you graph an inequality, you used a OPEN DOT when you use which symbols?

• $\le,\ge$

• $<,>$

Multiple Choice

Match the graph with its inequality.

• $x<-1$

• $x>-1$

• $x\le-1$

• $x\ge-1$

Multiple Choice

 $10\ge x-3$  

•  $x\le13$  

•  $x\ge13$  

•  $x\le7$  

•  $x\ge7$  

Multiple Choice

 $8x\ge-32$  

•  $x\le-4$  

•  $x\le4$  

•  $x\ge-4$  

•  $x\ge4$  

Multiple Choice

Consider this problem:

A fruit vendor can sell at most 100 apples and orange combined. How many of each kind can he choose?

Let x represent the number of apples, and y represent the number of oranges. What is the mathematical statement of the problem?

• x + y = 100

• x + y ≤ 100

Let's Explore :)

“WHAT AM I?”

Directions: Identify the situations which illustrate inequalities.

Multiple Choice

1. The value of one Philippine peso (p) is less than the value of one US dollar (d)

• Inequality

• Not

Multiple Choice

2. According to the NSO, there are more female (f) Filipinos than male (m) Filipinos

• Inequality

• Not

Multiple Choice

3. The number of girls (g) in the band is one more than twice the number of boys (b).

• Inequality

• Not

Multiple Choice

4. The school bus has a maximum seating capacity (c) of 80 persons

• Inequality

• Not

Multiple Choice

5. According to research, an average adult generates about 4 kg of waste daily (w)

• Inequality

• Not

Open Ended

What are the hint/s or word/s help you to decide whether the situation is an inequality or not? Can you write the inequality model of each situation?

Linear Inequalities in Two Variables

• An open sentence that makes use of the symbol "=" is called equation

• On the other hand, an open sentence that makes use of the relation symbols, < , > , ≤ , ≥ , and ≠ is called an inequality.

Linear Inequalities in Two Variables

• The general form of inequality in two variable is Ax + By > C, where A and B cannot be zero. The symbol > can be replaced by < , ≥ , or ≤.

• A solution of an equality in two variables is an ordered pair (x,y) whose coordinates satisfy the inequality.

Consider the inequality 2x - 5y < 4.

Let us check if the ordered pairs (-1, 2), (0, -3), and (3, 3) are solutions of the linear inequality.

To check if each pair of numbers satisfies the linear inequality, replace x and y with the x-coordinates and y-coordinates and see if the resulting inequality is true.

• Substituting (-1, 2) to the inequality, we have

  2x - 5y < 4

  2(-1) - 5(2) < 4

  -12 < 4 (TRUE)

  The ordered pair (-1, 2) is a solution of the inequality

• Substituting (0, -3) to the inequality, we have

  2x - 5y < 4

  2(0) - 5(-3) < 4

  15 < 4 (FALSE)

  The ordered pair (0, -3) is NOT a solution of the inequality

• Substituting (3, 3) to the inequality, we have

  2x - 5y < 4

  2(3) - 5(3) < 4

  -9 < 4 (TRUE)

  The ordered pair (3, 3) is a solution of the inequality

State whether each given ordered pair is a solution of the inequality.

Multiple Choice

 $2x–y>10;\ \ \ \ (7,2)$  

• Solution

• Not a solution

Multiple Choice

 $x+3y≤8;\ \ \ \ (4,-1)$  

• Solution

• Not a solution

Multiple Choice

 $7x–2y≥6;\ \ \ \ (-3,-8)$  

• Solution

• Not a solution

Multiple Choice

 $16–y>x;\ \ \ \ (-1,9)$  

• Solution

• Not Solution

Multiple Choice

 $y<4x–5;\ \ \ \ (0,0)$  

• Solution

• Not a solution

Graph of a Linear Inequality in Two Variables

The graph of a linear inequality in two variables is the set of all points in the rectangular coordinate system whose ordered pairs satisfy the inequality. When a line is graphed in the coordinate plane, it separates the plane into two regions called half- planes. The line that separates the plane is called the plane divider. 

Consider graphing the solution set of the inequality x + 4y > 12.

Step 1: Solve for the inequality for y.
 $x\ +\ 4y\ >\ 12$  
 $4y>-x+12$  

 $y>-\frac{1}{4}x+3$  

Step 2: Graph the boundary line

 $y=-\frac{1}{4}x+3$ . Use dashed line for >.

(NOTE: Use dashed line for  $>,<$   and solid line for  $\ge,\le$ .)

Step 3: Shade the appropriate half-plane

To decide on the appropriate half-plane, use a test point not on the line. If the test point satisfies the inequality, the region that contains the test point should be shaded. We can always use the origin (0,0) as a test point as long as the line does not pass through it. 

 $x+4y>12$  
 $0+4\left(0\right)>12$  
 $0>12$  false statement
A false statement means that the half-plane containing (0,0) should not be shaded. The origin (0,0), is not part of the solution set of the inequality 

Shade the part of the plane divider where the solutions of the inequality is found.

Multiple Choice

 $y<x+3$  

• right side of the plane divider

• left side of the plane divider

Multiple Choice

 $y–x>–5$  

• right side of the plane divider

• left side of the plane divider

Multiple Choice

 $x≤y–4$  

• right side of the plane divider

• left side of the plane divider

Multiple Choice

 $x+y≥1$  

• right side of the plane divider

• left side of the plane divider

Multiple Choice

 $2x+y<2$  

• right side of the plane divider

• left side of the plane divider