5.1 Q-1

Question Statement

Graph the solution set of the following linear inequalities in the $xy$-plane:

i. $2x + y \leq 6$

ii. $3x + 7y \geq 21$

iii. $3x - 2y \geq 6$

iv. $5x - 4y \leq 20$

v. $2x + 1 \geq 0$

vi. $3y - 4 \geq 0$

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Background and Explanation

To solve linear inequalities graphically, we first convert each inequality into an equation by replacing the inequality symbol with an equal sign. The equation represents a line on the $xy$-plane, and the inequality defines which side of the line is included in the solution set.

To determine which side of the line is the solution region, we test a point, typically $(0, 0)$, and check if it satisfies the inequality.

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Solution

i. $2x + y \leq 6$

1. Convert the inequality into an equation:
   $2x + y = 6$

2. Find two points on the line:

   • When $x = 0$, solve for $y$:
     $2(0) + y = 6 \quad \Rightarrow \quad y = 6 \quad \text{so the point is } (0, 6)$
   • When $y = 0$, solve for $x$:
     $2x + 0 = 6 \quad \Rightarrow \quad x = 3 \quad \text{so the point is } (3, 0)$

3. Test the inequality at $(0, 0)$:
   $2(0) + 0 = 0 \quad \Rightarrow \quad 0 \leq 6 \quad \text{which is true.}$
   So, the region below the line $2x + y = 6$ is the solution set.

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ii. $3x + 7y \geq 21$

1. Convert the inequality into an equation:
   $3x + 7y = 21$

2. Find two points on the line:

   • When $x = 0$, solve for $y$:
     $3(0) + 7y = 21 \quad \Rightarrow \quad y = 3 \quad \text{so the point is } (0, 3)$
   • When $y = 0$, solve for $x$:
     $3x + 0 = 21 \quad \Rightarrow \quad x = 7 \quad \text{so the point is } (7, 0)$

3. Test the inequality at $(0, 0)$:
   $3(0) + 7(0) = 0 \quad \Rightarrow \quad 0 \geq 21 \quad \text{which is false.}$
   So, the region above the line $3x + 7y = 21$ is the solution set.

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iii. $3x - 2y \geq 6$

1. Convert the inequality into an equation:
   $3x - 2y = 6$

2. Find two points on the line:

   • When $x = 0$, solve for $y$:
     $3(0) - 2y = 6 \quad \Rightarrow \quad -2y = 6 \quad \Rightarrow \quad y = -3 \quad \text{so the point is } (0, -3)$
   • When $y = 0$, solve for $x$:
     $3x - 0 = 6 \quad \Rightarrow \quad x = 2 \quad \text{so the point is } (2, 0)$

3. Test the inequality at $(0, 0)$:
   $3(0) - 2(0) = 0 \quad \Rightarrow \quad 0 \geq 6 \quad \text{which is false.}$
   So, the region above the line $3x - 2y = 6$ is the solution set.

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iv. $5x - 4y \leq 20$

1. Convert the inequality into an equation:
   $5x - 4y = 20$

2. Find two points on the line:

   • When $x = 0$, solve for $y$:
     $5(0) - 4y = 20 \quad \Rightarrow \quad -4y = 20 \quad \Rightarrow \quad y = -5 \quad \text{so the point is } (0, 5)$
   • When $y = 0$, solve for $x$:
     $5x - 0 = 20 \quad \Rightarrow \quad x = 4 \quad \text{so the point is } (4, 0)$

3. Test the inequality at $(0, 0)$:
   $5(0) - 4(0) = 0 \quad \Rightarrow \quad 0 \leq 20 \quad \text{which is true.}$
   So, the region below the line $5x - 4y = 20$ is the solution set.

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v. $2x + 1 \geq 0$

1. Convert the inequality into an equation:
   $2x + 1 = 0$

2. Solve for $x$:
   $2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}$

3. Test the inequality at $(0, 0)$:
   $2(0) + 1 = 1 \quad \Rightarrow \quad 1 \geq 0 \quad \text{which is true.}$
   So, the region to the right of the line $2x + 1 = 0$ is the solution set.

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vi. $3y - 4 \geq 0$

1. Convert the inequality into an equation:
   $3y - 4 = 0$

2. Solve for $y$:
   $3y = 4 \quad \Rightarrow \quad y = \frac{4}{3}$

3. Test the inequality at $(0, 0)$:
   $3(0) - 4 = -4 \quad \Rightarrow \quad -4 \geq 0 \quad \text{which is false.}$
   So, the region above the line $3y - 4 = 0$ is the solution set.

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Key Formulas or Methods Used

• Linear inequality: When solving linear inequalities, first convert the inequality into an equation to graph the boundary line.
• Test point method: After plotting the boundary line, use a test point (typically $(0, 0)$) to determine which side of the line satisfies the inequality.
• Solution region: Depending on whether the inequality is $\leq$, $\geq$, $<$, or $>$, shade the appropriate region of the plane.

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Summary of Steps

1. Convert each inequality into an equation.
2. Find two points that lie on the line.
3. Plot these points and draw the line.
4. Test the inequality using a point, usually $(0, 0)$, to determine the region that satisfies the inequality.
5. Shade the appropriate region based on the inequality symbol.