Lesson Objectives

Dear learner;

At the end of the lesson you will be able to:

• Graph linear inequalities in two variables on a coordinate plane to visualize the solution set.
• Determine the shading region by analyzing the inequality symbol.
• Test a point not on the boundary line to verify the accuracy of the shaded region in a graphical representation of linear inequalities in two variables.

Key Terms

• system of linear inequalites
• solution
• graphical solution
• shaded region

Brainstorming questions

1. A school in Addis Ababa is planning to purchase notebooks and pens for its students. The school has a budget of 8,000 Ethiopian Birr. Each notebook costs 120 Ethiopian Birr and each pen costs 20 Ethiopian Birr. If the school wants to buy at least 30 notebooks, how many pens can it buy while staying within the budget?

Solution:
To solve this, define:

• ( x ) as the number of notebooks.
• ( y ) as the number of pens.

The inequalities representing the constraints are:

1. Budget constraint: $( 120x + 20y \leq 8000 )$
2. Minimum number of notebooks: $( x \geq 30 )$

Substitute ( x ) with 30 in the budget constraint:
$[ 120(30) + 20y \leq 8000 ]$
$[ 3600 + 20y \leq 8000 ]$
$[ 20y \leq 4400 ]$
$[ y \leq 220 ]$

Therefore, the school can buy up to 220 pens if it purchases at least 30 notebooks while staying within the budget.

A vendor in Addis Ababa rents a market stall for 1,500 Ethiopian Birr per month and sells two types of fruits: bananas and apples. Each banana sells for 10 Ethiopian Birr, and each apple sells for 15 Ethiopian Birr. The vendor needs to cover the rent and make a profit of at least 2,000 Ethiopian Birr in total. How many bananas and apples must the vendor sell to achieve this goal?2.

Solution:
Define:

• ( x ) as the number of bananas sold.
• ( y ) as the number of apples sold.

The inequalities are:

1. Revenue needed to cover rent and achieve profit: $( 10x + 15y \geq 3500 ) (1,500 Birr for rent + 2,000 Birr profit)$
2. Non-negativity constraints: $( x \geq 0 ) and ( y \geq 0 )$

Graphically solve this by plotting:
$[ 10x + 15y \geq 3500 ]$

The solution region represents all possible combinations of bananas and apples that meet or exceed the revenue goal.

2.1. Systems of Linear Inequalities in Two Variables

Definition: Systems of Linear Inequalities in Two Variables:

A set of two or more linear inequalities with two variables, (x) and (y). written of the form:

$\begin{align’} ax + by & \leq c \\ dx + ky & \geq q \end{align’}$

where a,b,c,d,k and q are real numbers. The solution is the region where the shaded areas of all inequalities overlap on the coordinate plane.

• Linear inequalities in two variables describe the unequal relationship between two algebraic expressions containing two separate variable
• Linear inequalities in two variables illustrate the disparity between two algebraic expressions involving two different variables.
• The unequal comparison between two distinct variables is expressed through linear inequalities in two variables.
• Linear inequalities in two variables showcase the unequal correlation between two algebraic expressions with two distinct variables.

When solving inequalities in two variables graphically follow these key procedures:

1. Plot the inequality on a coordinate plane: Begin by graphing the inequality on a coordinate plane by identifying the boundary line (usually change inequalities in to equal sign to plot the boundary line). Use a dashed/broken line for strict inequalities (<, >) and a solid line for non-strict inequalities (≤, ≥).
2. Determine the shading region: Next, determine which side of the boundary line to shade based on the inequality symbol. If the inequality is greater than or less than, shade the region above or below the line, respectively. If the inequality is greater than or equal to or less than or equal to, include the boundary line in the shaded region.
3. Test a point: To confirm the correct shaded region, choose a test point not on the boundary line and substitute its coordinates into the original inequality. If the inequality holds true for the test point, then the shaded region is correct. If not, adjust the shading accordingly.
4. Label the solution region: Finally, label the shaded region as the solution set for the inequality. This region represents all the points that satisfy the given inequality in two variables.

Example 1: Solve the following linear inequalities in two variables using graphical method.

a) 2x -𝑦 > 6

Fig 1: graph of 2x-y=6

fig 2: graph of 2x – y > 6

b) y > x + 1

solution:

step1: Draw the graph of y = x+1 with a broken line

step 2: determine the solution region using a point out of y = x + 1.

take p(0,0): and substitute ion the inequalty y > x+1 .

substitution results 0 > 0 + 1 implies 0 > 1 is false so the region is to the opposite side of the point (0, 0).

finaly Draw the graph and shade the region.

fig 3: graph of y > x +1

Example 2: solve the system :

$\begin{align’} 2x – y & \leq 3\\x + 4y & \geq 2 \end{align’}$

Solution:

step 1: draw the graph of 2x – y = 3 and x + 4y = 2 on the coordinate plane

step2: determine the region of each inequality using a point p(0, 0)

substitute p(0, 0) in $2x – y \leq 3 $ and $x + 4y \geq 2$ respectively such that $0\leq3$ is true and the region is along the point p(0, 0)

on the second equation $0\geq2$ is false so that the region lies to the opposite of the point p(0, 0)

Draw the graph: first $2x – y \leq 3 $

then add graph of $x + 4y \geq 2$:

Here it shows that the common region is the solution for the system of linear inequalities

finally put the solution region only by erasing other than the common region

Example 3: There are at most 56 people composed of children and adults who are in a bus. Each child and adult paid birr 80 and birr 100, respectively. If the total amount collected was not more than birr 4,800, how many children and adults are in the bus?

Solution: Let x = number of children in the bus

                      y = number of adults in the bus

Represent the number of people in the bus as x + y ≤ 56.

Represent the amount collected as 80x + 100y ≤ 4,800

Use the two inequalities to find the number of children and adults who are in the bus. Write these as a system of linear inequalities then solve graphically.