Which Ordered Pairs Make Both Inequalities True? Check All That Apply.A. { (-5, 5)$}$B. { (0, 3)$}$C. { (0, -2)$}$D. { (1, 1)$}$E. { (3, -4)$}$
Introduction
In mathematics, inequalities are used to compare two or more values. They are an essential part of algebra and are used to solve a wide range of problems. In this article, we will focus on solving inequalities and finding the ordered pairs that make both inequalities true.
What are Inequalities?
Inequalities are mathematical statements that compare two or more values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. For example:
- 2x + 3 > 5
- x - 2 < 3
- 4x ≥ 12
- x + 2 ≤ 5
Solving Inequalities
To solve an inequality, we need to isolate the variable (x) 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.
Step 1: Simplify the Inequality
The first step in solving an inequality is to simplify it by combining like terms and removing any parentheses.
Step 2: Add or Subtract the Same Value
Next, we need to add or subtract the same value to both sides of the inequality to isolate the variable.
Step 3: Multiply or Divide Both Sides
Finally, we need to multiply or divide both sides of the inequality by the same value to isolate the variable.
Example 1: Solving the Inequality 2x + 3 > 5
To solve the inequality 2x + 3 > 5, we need to isolate the variable x.
- Subtract 3 from both sides: 2x > 5 - 3
- Simplify: 2x > 2
- Divide both sides by 2: x > 1
Example 2: Solving the Inequality x - 2 < 3
To solve the inequality x - 2 < 3, we need to isolate the variable x.
- Add 2 to both sides: x < 3 + 2
- Simplify: x < 5
Example 3: Solving the Inequality 4x ≥ 12
To solve the inequality 4x ≥ 12, we need to isolate the variable x.
- Divide both sides by 4: x ≥ 12/4
- Simplify: x ≥ 3
Example 4: Solving the Inequality x + 2 ≤ 5
To solve the inequality x + 2 ≤ 5, we need to isolate the variable x.
- Subtract 2 from both sides: x ≤ 5 - 2
- Simplify: x ≤ 3
Which Ordered Pairs Make Both Inequalities True?
Now that we have solved the inequalities, we need to find the ordered pairs that make both inequalities true. We will use the following inequalities:
A. 2x + 3 > 5 B. x - 2 < 3 C. 4x ≥ 12 D. x + 2 ≤ 5
Option A: 2x + 3 > 5
To find the ordered pairs that make both inequalities true, we need to solve the inequality 2x + 3 > 5.
- Subtract 3 from both sides: 2x > 5 - 3
- Simplify: 2x > 2
- Divide both sides by 2: x > 1
The ordered pairs that make the inequality 2x + 3 > 5 true are:
- (2, 3)
- (3, 4)
- (4, 5)
- ...
Option B: x - 2 < 3
To find the ordered pairs that make both inequalities true, we need to solve the inequality x - 2 < 3.
- Add 2 to both sides: x < 3 + 2
- Simplify: x < 5
The ordered pairs that make the inequality x - 2 < 3 true are:
- (1, 2)
- (2, 3)
- (3, 4)
- ...
Option C: 4x ≥ 12
To find the ordered pairs that make both inequalities true, we need to solve the inequality 4x ≥ 12.
- Divide both sides by 4: x ≥ 12/4
- Simplify: x ≥ 3
The ordered pairs that make the inequality 4x ≥ 12 true are:
- (3, 4)
- (4, 5)
- (5, 6)
- ...
Option D: x + 2 ≤ 5
To find the ordered pairs that make both inequalities true, we need to solve the inequality x + 2 ≤ 5.
- Subtract 2 from both sides: x ≤ 5 - 2
- Simplify: x ≤ 3
The ordered pairs that make the inequality x + 2 ≤ 5 true are:
- (1, 2)
- (2, 3)
- (3, 4)
- ...
Conclusion
In this article, we have learned how to solve inequalities and find the ordered pairs that make both inequalities true. We have used the following inequalities:
A. 2x + 3 > 5 B. x - 2 < 3 C. 4x ≥ 12 D. x + 2 ≤ 5
We have solved each inequality and found the ordered pairs that make both inequalities true. We have also discussed the importance of solving inequalities and finding the ordered pairs that make both inequalities true.
Which Ordered Pairs Make Both Inequalities True? Check All That Apply
Based on the inequalities we have solved, the ordered pairs that make both inequalities true are:
- (2, 3)
- (3, 4)
- (4, 5)
- (1, 2)
- (2, 3)
- (3, 4)
Therefore, the correct answer is:
A. {(-5, 5)$}$ B. {(0, 3)$}$ C. {(0, -2)$}$ D. {(1, 1)$}$ E. {(3, -4)$}$
Q: What is an inequality?
A: An inequality is a mathematical statement that compares two or more values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate the variable (x) 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 an inequality and an equation?
A: An equation is a mathematical statement that states that two or more values are equal. An inequality, on the other hand, states that two or more values are not equal.
Q: Can I use the same methods to solve inequalities as I do to solve equations?
A: Yes, you can use the same methods to solve inequalities as you do to solve equations. However, you need to be careful when multiplying or dividing both sides of an inequality by a negative number, as this can change the direction of the inequality.
Q: How do I know which direction to change the inequality when multiplying or dividing by a negative number?
A: When multiplying or dividing both sides of an inequality by a negative number, you need to change the direction of the inequality. For example, if you have the inequality x > 2 and you multiply both sides by -1, the inequality becomes x < -2.
Q: Can I use inequalities to solve real-world problems?
A: Yes, inequalities can be used to solve real-world problems. For example, you can use inequalities to determine the maximum or minimum value of a quantity, or to compare the values of two or more quantities.
Q: How do I graph an inequality on a number line?
A: To graph an inequality on a number line, you need to plot a point on the number line that represents the value of the variable (x). Then, you need to shade the region of the number line that represents the solution to the inequality.
Q: Can I use inequalities to solve systems of equations?
A: Yes, inequalities can be used to solve systems of equations. For example, you can use inequalities to find the intersection of two or more lines, or to determine the maximum or minimum value of a quantity.
Q: How do I use inequalities to solve systems of equations?
A: To use inequalities to solve systems of equations, you need to first solve each equation separately. Then, you need to use the inequalities to find the intersection of the two or more lines.
Q: Can I use inequalities to solve optimization problems?
A: Yes, inequalities can be used to solve optimization problems. For example, you can use inequalities to find the maximum or minimum value of a quantity, or to determine the optimal value of a variable.
Q: How do I use inequalities to solve optimization problems?
A: To use inequalities to solve optimization problems, you need to first define the objective function and the constraints of the problem. Then, you need to use the inequalities to find the optimal value of the variable.
Conclusion
In this article, we have answered some of the most frequently asked questions about inequalities. We have discussed the basics of inequalities, how to solve them, and how to use them to solve real-world problems. We have also discussed how to graph inequalities on a number line and how to use inequalities to solve systems of equations and optimization problems.