Inequalities

1. First constraint: x, y ≥ 0
   This signifies that the region lies in the first quadrant where both x and y are non-negative.
2. Second constraint: y ≤ 8
   This implies a horizontal line at y = 8 and the region is below this line.
3. Third constraint: x + y ≤ 4
   This is the equation of a line that intersects the axes. Setting x = 0 gives y = 4 and setting y = 0 gives x = 4. Therefore, the line connects the points (0, 4) and (4, 0).

• The intersection of y ≤ 8 and x + y ≤ 4 lies within a triangle formed by the points (0, 0), (0, 4), and (4, 0).

Given inequality:


Case 1: If , then
In this case, the inequality becomes:
, which is not true (equality, not strict inequality).

Case 2: If , then
So the inequality becomes:


Conclusion:
The inequality is satisfied when

Final Answer:

To determine the solution region of the inequality , follow these steps:

1. First, identify the boundary line associated with the inequality, which is .
2. Simplify the equation by dividing every term by 2: .
3. Plot the line on the coordinate plane.
4. Determine if the inequality is less than or equal ( ) by substituting the origin into the inequality:

(True)

1. Since the inequality holds true for the origin, the region containing the origin is part of the solution.
2. Since the inequality is , it includes the boundary line, indicating a closed half-plane.

Thus, the solution region is a closed half-plane containing the origin.

To solve the inequality , we need to consider the properties of absolute values. The absolute value of a number , , can be defined as:

• if
• if

Let's consider the expression and separately:

• if and if
• if and if

Next, we solve for the critical points that determine intervals for . Set , solving gives .

We analyze the inequality in three intervals determined by , :

• Interval 1:
• Interval 2:
• Interval 3:

Evaluate the inequality in each interval:

• For : We have . This simplifies to or . Further simplifying gives or . But this is not possible for .
• For : We have . Simplifying gives or . This means .
• For : We have . Simplifying gives , which is always true for .

Combine the solutions from the intervals. The overall solution set is .

Thus, the solution set of the inequality is .

Step 1: Understanding the Concept:

This question requires evaluating four separate mathematical statements (inequalities and equations) to determine their validity.

Step 2: Detailed Explanation:

Statement (A):

Squaring both sides to compare:

LHS squared:

RHS squared:

We get , and since , this statement is true.

Statement (B): If and , then

Multiplying or dividing by a negative number reverses the inequality. This is a fundamental rule of inequalities, so this statement is true.

Statement (C): , if

For , we find that . This confirms the inequality, so this statement is true.

Statement (D): If and are positive integers and , then

Solving the equation, we find , so . Therefore, this statement is false.

Step 3: Final Answer:

Statements (A), (B), and (C) are true, while (D) is false. The correct choice is the one that lists (A), (B), and (C) only.

Step 1: Understanding the Concept:

This question requires evaluating four separate mathematical statements (inequalities and equations) to determine their validity.

Step 2: Detailed Explanation:

Statement (A):

Squaring both sides to compare:

LHS squared:

RHS squared:

We get , and since , this statement is true.

Statement (B): If and , then

Multiplying or dividing by a negative number reverses the inequality. This is a fundamental rule of inequalities, so this statement is true.

Statement (C): , if

For , we find that . This confirms the inequality, so this statement is true.

Statement (D): If and are positive integers and , then

Solving the equation, we find , so . Therefore, this statement is false.

Step 3: Final Answer:

Statements (A), (B), and (C) are true, while (D) is false. The correct choice is the one that lists (A), (B), and (C) only.