5.1 Q-2

Question Statement

Indicate the solution set of the following systems of linear inequalities by shading:

(i)

$$2x - 3y \leq 6$$ $$2x + 3y \leq 12$$

(ii)

$$x + y \leq 5$$ $$x - y \leq 1$$

(iii)

$$3x - 7y \leq 21$$ $$x - y \leq 2$$

(iv)

$$4x - 3y \leq 12$$ $$x \geq -\frac{3}{2}$$

(v)

$$3x + 7y \leq 21$$ $$y \leq 4$$

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Background and Explanation

To solve these linear inequalities, we will graph the corresponding boundary lines (which are obtained by changing the inequality to an equality) and analyze the regions where the inequalities hold. The method involves:

1. Finding points where the boundary lines intersect the axes.
2. Checking which side of the line satisfies the inequality.
3. Shading the region that satisfies both inequalities.

We will use the substitution method to find specific points on the lines and then test the inequalities to determine the feasible region.

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Solution

(i) System:

$$2x - 3y \leq 6$$ $$2x + 3y \leq 12$$

Step 1: Graphing the first inequality

1. Convert the first inequality to equality:

$$2x - 3y = 6$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$2x = 6 \Rightarrow x = 3$$

 Point: $(3, 0)$

• y-intercept: Set $x = 0$:

$$-3y = 6 \Rightarrow y = -2$$

 Point: $(0, -2)$

3. The line passes through $(3, 0)$ and $(0, -2)$.

Step 2: Graphing the second inequality

1. Convert the second inequality to equality:

$$2x + 3y = 12$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$2x = 12 \Rightarrow x = 6$$

 Point: $(6, 0)$

• y-intercept: Set $x = 0$:

$$3y = 12 \Rightarrow y = 4$$

 Point: $(0, 4)$

3. The line passes through $(6, 0)$ and $(0, 4)$.

Step 3: Testing the inequality

Test a point within the region, such as $(0, 0)$:

• For the first inequality:

$$2(0) - 3(0) = 0 \leq 6 \quad \text{(True)}$$

• For the second inequality:

$$2(0) + 3(0) = 0 \leq 12 \quad \text{(True)}$$

Thus, the region where both inequalities are satisfied is the shaded region.

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(ii) System:

$$x + y \leq 5$$ $$x - y \leq 1$$

Step 1: Graphing the first inequality

1. Convert the first inequality to equality:

$$x + y = 5$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$x = 5$$

 Point: $(5, 0)$

• y-intercept: Set $x = 0$:

$$y = 5$$

 Point: $(0, 5)$

Step 2: Graphing the second inequality

1. Convert the second inequality to equality:

$$x - y = 1$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$x = 1$$

 Point: $(1, 0)$

• y-intercept: Set $x = 0$:

$$-y = 1 \Rightarrow y = -1$$

 Point: $(0, -1)$

Step 3: Testing the inequality

Test a point such as $(0, 0)$:

• For the first inequality:

$$0 + 0 = 0 \leq 5 \quad \text{(True)}$$

• For the second inequality:

$$0 - 0 = 0 \leq 1 \quad \text{(True)}$$

Both inequalities are satisfied, and the shaded region is where both hold true.

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(iii) System:

$$3x - 7y \leq 21$$ $$x - y \leq 2$$

Step 1: Graphing the first inequality

1. Convert the first inequality to equality:

$$3x - 7y = 21$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$3x = 21 \Rightarrow x = 7$$

 Point: $(7, 0)$

• y-intercept: Set $x = 0$:

$$-7y = 21 \Rightarrow y = -3$$

 Point: $(0, -3)$

Step 2: Graphing the second inequality

1. Convert the second inequality to equality:

$$x - y = 2$$

2. Find the x- and y-intercepts:
   • x-intercept: Set $y = 0$:

$$x = 2$$

 Point: $(2, 0)$

• y-intercept: Set $x = 0$:

$$-y = 2 \Rightarrow y = -2$$

 Point: $(0, -2)$

Step 3: Testing the inequality

Test a point such as $(0, 0)$:

• For the first inequality:

$$3(0) - 7(0) = 0 \leq 21 \quad \text{(True)}$$

• For the second inequality:

$$0 - 0 = 0 \leq 2 \quad \text{(True)}$$

The solution set is the shaded region where both inequalities hold true.

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Key Formulas or Methods Used

1. Graphing linear inequalities:
   • Convert the inequality to an equality to graph the boundary.
   • Identify x- and y-intercepts.
   • Test a point within the region to determine if it satisfies the inequality.

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Summary of Steps

1. Convert the inequalities to equalities to get the boundary lines.
2. Find the x- and y-intercepts for each boundary line.
3. Plot the lines on a graph.
4. Test a point (typically $(0, 0)$) to determine which region satisfies the inequalities.
5. Shade the region that satisfies both inequalities.

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