Which Ordered Pairs Make Both Inequalities True? Check All That Apply.A. { (-2, 2)$}$ B. { (0, 0)$}$ C. { (1, 1)$}$ D. { (1, 3)$}$ E. { (2, 2)$}$
Introduction
In mathematics, inequalities are used to compare values and express relationships between variables. When solving inequalities, it's essential to find the ordered pairs that satisfy both inequalities. In this article, we will explore how to solve inequalities and check which ordered pairs make both inequalities true.
Understanding Inequalities
Inequalities are mathematical statements that compare two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). For example:
- 2x + 3 > 5
- x - 2 < 3
- 4x ≥ 12
- x + 2 ≤ 7
Solving Inequalities
To solve inequalities, we need to isolate the variable on one side of the inequality sign. We can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Adding and Subtracting
When adding or subtracting the same value to both sides of an inequality, the inequality sign remains the same.
- 2x + 3 > 5
- 2x > 5 - 3
- 2x > 2
- x > 1
Multiplying and Dividing
When multiplying or dividing both sides of an inequality by a negative value, the inequality sign is reversed.
- 2x > 4
- x > 4/2
- x > 2
- 2x - 4 < 0
- 2x < 4
- x < 2
Solving Linear Inequalities
Linear inequalities are inequalities that can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants.
- 2x + 3 > 5
- 2x > 5 - 3
- 2x > 2
- x > 1
- 2x - 4 < 0
- 2x < 4
- x < 2
Solving Quadratic Inequalities
Quadratic inequalities are inequalities that can be written in the form ax^2 + bx + c > 0, ax^2 + bx + c < 0, ax^2 + bx + c ≥ 0, or ax^2 + bx + c ≤ 0, where a, b, and c are constants.
- x^2 + 4x + 4 > 0
- (x + 2)^2 > 0
- x + 2 > 0
- x > -2
Solving Absolute Value Inequalities
Absolute value inequalities are inequalities that involve absolute value expressions.
- |x| > 2
- x > 2 or x < -2
- |x + 2| > 0
- x + 2 > 0 or x + 2 < 0
- x > -2 or x < -4
Checking Ordered Pairs
Now that we have solved the inequalities, let's check which ordered pairs make both inequalities true.
Checking Ordered Pair A
Ordered pair A is (-2, 2). Let's substitute x = -2 and y = 2 into the inequalities.
- 2x + 3 > 5
- 2(-2) + 3 > 5
- -4 + 3 > 5
- -1 > 5 (False)
Ordered pair A does not make both inequalities true.
Checking Ordered Pair B
Ordered pair B is (0, 0). Let's substitute x = 0 and y = 0 into the inequalities.
- 2x + 3 > 5
- 2(0) + 3 > 5
- 3 > 5 (False)
Ordered pair B does not make both inequalities true.
Checking Ordered Pair C
Ordered pair C is (1, 1). Let's substitute x = 1 and y = 1 into the inequalities.
- 2x + 3 > 5
- 2(1) + 3 > 5
- 5 > 5 (False)
Ordered pair C does not make both inequalities true.
Checking Ordered Pair D
Ordered pair D is (1, 3). Let's substitute x = 1 and y = 3 into the inequalities.
- 2x + 3 > 5
- 2(1) + 3 > 5
- 5 > 5 (False)
Ordered pair D does not make both inequalities true.
Checking Ordered Pair E
Ordered pair E is (2, 2). Let's substitute x = 2 and y = 2 into the inequalities.
- 2x + 3 > 5
- 2(2) + 3 > 5
- 7 > 5 (True)
Ordered pair E makes both inequalities true.
Conclusion
Q: What is an inequality?
A: An inequality is a mathematical statement that compares two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Q: What is the difference between a linear inequality and a quadratic inequality?
A: A linear inequality is an inequality that can be written in the form ax + b > c, ax + b < c, ax + b ≥ c, or ax + b ≤ c, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c > 0, ax^2 + bx + c < 0, ax^2 + bx + c ≥ 0, or ax^2 + bx + c ≤ 0, where a, b, and c are constants.
Q: How do I solve an absolute value inequality?
A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive, and when it is negative. You can then solve each case separately and combine the solutions.
Q: What is the order of operations for solving inequalities?
A: The order of operations for solving inequalities is the same as for solving equations: parentheses, exponents, multiplication and division, and addition and subtraction.
Q: Can I use the same steps to solve all types of inequalities?
A: No, the steps for solving inequalities may vary depending on the type of inequality. For example, solving a quadratic inequality may require factoring or using the quadratic formula, while solving an absolute value inequality may require considering two cases.
Q: How do I check which ordered pairs make both inequalities true?
A: To check which ordered pairs make both inequalities true, you need to substitute the values of the ordered pair into both inequalities and check if the inequalities are true.
Q: What if I get a false statement when checking an ordered pair?
A: If you get a false statement when checking an ordered pair, it means that the ordered pair does not make both inequalities true.
Q: Can I use a graphing calculator to solve inequalities?
A: Yes, you can use a graphing calculator to solve inequalities. Graphing calculators can help you visualize the solution set of an inequality and find the ordered pairs that make both inequalities true.
Q: How do I know if an inequality is true or false?
A: To determine if an inequality is true or false, you need to substitute a value into the inequality and check if the inequality is true or false.
Q: Can I use inequalities to solve real-world problems?
A: Yes, inequalities can be used to solve real-world problems. Inequalities can be used to model real-world situations, such as comparing the cost of two products or determining the maximum or minimum value of a quantity.
Q: How do I apply inequalities to real-world problems?
A: To apply inequalities to real-world problems, you need to identify the variables and constants in the problem, write an inequality to model the situation, and solve the inequality to find the solution.
Q: What are some common applications of inequalities?
A: Some common applications of inequalities include:
- Comparing the cost of two products
- Determining the maximum or minimum value of a quantity
- Modeling real-world situations, such as population growth or financial planning
- Solving optimization problems, such as finding the maximum or minimum value of a function
Conclusion
In this article, we have answered some frequently asked questions about solving inequalities. We have covered topics such as the order of operations, solving linear and quadratic inequalities, and checking which ordered pairs make both inequalities true. We have also discussed the applications of inequalities in real-world problems and how to apply inequalities to solve optimization problems.