Match Each Inequality To Its Solution.a. B. C. D. 1. $-3x \ \textgreater \ -36$ 2. $b + 5 \ \textgreater \ 23$ 3. $1 + 7n \geq -90$ 4. $\frac{x}{2} - 2 \ \textgreater \ 1$
Introduction
Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will explore how to solve four different inequalities and match each inequality to its solution.
Inequality 1: -3x > -36
Step 1: Divide both sides by -3
To solve the inequality -3x > -36, we need to isolate the variable x. We can do this by dividing both sides of the inequality by -3.
\frac{-3x}{-3} > \frac{-36}{-3}
Step 2: Simplify the inequality
When we divide both sides of the inequality by -3, we need to remember that the inequality sign will change direction because we are dividing by a negative number.
x < 12
Solution
The solution to the inequality -3x > -36 is x < 12.
Inequality 2: b + 5 > 23
Step 1: Subtract 5 from both sides
To solve the inequality b + 5 > 23, we need to isolate the variable b. We can do this by subtracting 5 from both sides of the inequality.
b + 5 - 5 > 23 - 5
Step 2: Simplify the inequality
When we subtract 5 from both sides of the inequality, we get:
b > 18
Solution
The solution to the inequality b + 5 > 23 is b > 18.
Inequality 3: 1 + 7n ≥ -90
Step 1: Subtract 1 from both sides
To solve the inequality 1 + 7n ≥ -90, we need to isolate the variable n. We can do this by subtracting 1 from both sides of the inequality.
1 + 7n - 1 ≥ -90 - 1
Step 2: Simplify the inequality
When we subtract 1 from both sides of the inequality, we get:
7n ≥ -91
Step 3: Divide both sides by 7
To solve the inequality 7n ≥ -91, we need to isolate the variable n. We can do this by dividing both sides of the inequality by 7.
\frac{7n}{7} ≥ \frac{-91}{7}
Step 4: Simplify the inequality
When we divide both sides of the inequality by 7, we get:
n ≥ -13
Solution
The solution to the inequality 1 + 7n ≥ -90 is n ≥ -13.
Inequality 4: x/2 - 2 > 1
Step 1: Add 2 to both sides
To solve the inequality x/2 - 2 > 1, we need to isolate the variable x. We can do this by adding 2 to both sides of the inequality.
x/2 - 2 + 2 > 1 + 2
Step 2: Simplify the inequality
When we add 2 to both sides of the inequality, we get:
x/2 > 3
Step 3: Multiply both sides by 2
To solve the inequality x/2 > 3, we need to isolate the variable x. We can do this by multiplying both sides of the inequality by 2.
\frac{x}{2} \cdot 2 > 3 \cdot 2
Step 4: Simplify the inequality
When we multiply both sides of the inequality by 2, we get:
x > 6
Solution
The solution to the inequality x/2 - 2 > 1 is x > 6.
Conclusion
In this article, we have explored how to solve four different inequalities and match each inequality to its solution. We have used step-by-step instructions to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality. By following these steps, you can solve any inequality and find its solution.
Matching Inequalities to Their Solutions
| Inequality | Solution |
|---|---|
| -3x > -36 | x < 12 |
| b + 5 > 23 | b > 18 |
| 1 + 7n ≥ -90 | n ≥ -13 |
| x/2 - 2 > 1 | x > 6 |
Q: What is an inequality?
A: An inequality is a mathematical expression that compares two values, often 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 on one side of the inequality sign, while maintaining the direction of the inequality. This can be done 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 expression that states that two values are equal, using an equals sign (=). An inequality, on the other hand, states that two values are not equal, using a greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbol.
Q: How do I know which direction to change the inequality sign when I multiply or divide both sides?
A: When you multiply or divide both sides of an inequality by a negative number, you need to change the direction of the inequality sign. For example, if you have the inequality x > 5 and you multiply both sides by -2, the inequality sign would change to x < -10.
Q: Can I add or subtract the same value to both sides of an inequality?
A: Yes, you can add or subtract the same value to both sides of an inequality. This will not change the direction of the inequality sign.
Q: Can I multiply or divide both sides of an inequality by the same value?
A: Yes, you can multiply or divide both sides of an inequality by the same value. However, if you multiply or divide both sides by a negative number, you need to change the direction of the inequality sign.
Q: How do I write the solution to an inequality in interval notation?
A: To write the solution to an inequality in interval notation, you need to represent the variable as a letter (e.g., x, b, n) and use the following notation:
- (a, b) represents the interval between a and b, but not including a and b.
- [a, b] represents the interval between a and b, including a and b.
- (-∞, a) represents the interval from negative infinity to a.
- (a, ∞) represents the interval from a to positive infinity.
- (-∞, a] represents the interval from negative infinity to a, including a.
- [a, ∞) represents the interval from a to positive infinity, including a.
Q: What is the difference between a strict inequality and a non-strict inequality?
A: A strict inequality is an inequality that uses a greater than (>), less than (<), or strict greater than or equal to (>) symbol. A non-strict inequality is an inequality that uses a greater than or equal to (≥) or less than or equal to (≤) symbol.
Q: Can I solve an inequality with multiple variables?
A: Yes, you can solve an inequality with multiple variables. However, you need to isolate one variable on one side of the inequality sign, while maintaining the direction of the inequality.
Q: How do I check my solution to an inequality?
A: To check your solution to an inequality, you need to plug in a value from the solution set into the original inequality and make sure it is true.
Q: What are some common mistakes to avoid when solving inequalities?
A: Some common mistakes to avoid when solving inequalities include:
- Changing the direction of the inequality sign when multiplying or dividing both sides by a negative number.
- Adding or subtracting the same value to both sides of an inequality without changing the direction of the inequality sign.
- Multiplying or dividing both sides of an inequality by a negative number without changing the direction of the inequality sign.
- Not isolating the variable on one side of the inequality sign.
- Not maintaining the direction of the inequality sign when solving the inequality.