Select The Values That Make The Inequality 8 Q \textless 72 8q \ \textless \ 72 8 Q \textless 72 True. Then Write An Equivalent Inequality In Terms Of Q Q Q .Values: 4, 6, 8, 9, 10Equivalent Inequality: Q \textless 9 Q \ \textless \ 9 Q \textless 9

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 finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $8q < 72$ and writing an equivalent inequality in terms of $q$.

Understanding the Inequality

The given inequality is $8q < 72$. This means that the product of $8$ and $q$ must be less than $72$. To solve this inequality, we need to isolate the variable $q$.

Step 1: Divide Both Sides by 8

To isolate $q$, we can divide both sides of the inequality by $8$. This will give us:

$$\frac{8q}{8} < \frac{72}{8}$$

Simplifying both sides, we get:

$$q < 9$$

Step 2: Write the Equivalent Inequality

Now that we have isolated $q$, we can write the equivalent inequality in terms of $q$. The equivalent inequality is:

$$q < 9$$

Selecting the Values

We are given the values $4, 6, 8, 9, 10$. To determine which values make the inequality $q < 9$ true, we can substitute each value into the inequality.

• For $q = 4$, we have $4 < 9$, which is true.
• For $q = 6$, we have $6 < 9$, which is true.
• For $q = 8$, we have $8 < 9$, which is true.
• For $q = 9$, we have $9 < 9$, which is false.
• For $q = 10$, we have $10 < 9$, which is false.

Therefore, the values that make the inequality $q < 9$ true are $4, 6, 8$.

Conclusion

In this article, we solved the inequality $8q < 72$ and wrote an equivalent inequality in terms of $q$. We also selected the values that make the inequality true and found that the values $4, 6, 8$ satisfy the inequality.

Key Takeaways

• To solve an inequality, we need to isolate the variable.
• Dividing both sides of the inequality by a constant can help us isolate the variable.
• The equivalent inequality is a rewritten version of the original inequality in terms of the variable.
• Selecting the values that make the inequality true involves substituting each value into the inequality.

Real-World Applications

Solving inequalities has many real-world applications, such as:

• Finance: Inequalities can be used to compare the growth rates of different investments.
• Science: Inequalities can be used to model the behavior of physical systems, such as the motion of objects.
• Engineering: Inequalities can be used to design and optimize systems, such as bridges and buildings.

Common Mistakes

When solving inequalities, it's common to make mistakes such as:

• Not isolating the variable.
• Not checking the direction of the inequality.
• Not considering the domain of the variable.

Tips and Tricks

To solve inequalities effectively, here are some tips and tricks:

• Use inverse operations to isolate the variable.
• Check the direction of the inequality by substituting a value into the inequality.
• Consider the domain of the variable to ensure that the solution is valid.

Conclusion

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 by using inverse operations. This involves adding, subtracting, multiplying, or dividing both sides of the inequality by a constant to get the variable on one side.

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, while an inequality states that two values are not equal. For example, the equation 2x = 4 is different from the inequality 2x < 4.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you need to check the sign of the coefficient of the variable. If the coefficient is positive, the inequality is either ≥ or ≤. If the coefficient is negative, the inequality is either < or >.

Q: What is the equivalent inequality?

A: The equivalent inequality is a rewritten version of the original inequality in terms of the variable. It is obtained by isolating the variable and changing the direction of the inequality if necessary.

Q: How do I select the values that make the inequality true?

A: To select the values that make the inequality true, you need to substitute each value into the inequality and check if it satisfies the inequality.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

• Not isolating the variable
• Not checking the direction of the inequality
• Not considering the domain of the variable

Q: How do I use inequalities in real-world applications?

A: Inequalities have many real-world applications, such as:

• Finance: Inequalities can be used to compare the growth rates of different investments.
• Science: Inequalities can be used to model the behavior of physical systems, such as the motion of objects.
• Engineering: Inequalities can be used to design and optimize systems, such as bridges and buildings.

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include:

• Using inverse operations to isolate the variable
• Checking the direction of the inequality by substituting a value into the inequality
• Considering the domain of the variable to ensure that the solution is valid

Q: How do I check if my solution is valid?

A: To check if your solution is valid, you need to substitute the solution into the original inequality and check if it satisfies the inequality.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: These are inequalities that involve a linear expression, such as 2x + 3 < 5.
• Quadratic inequalities: These are inequalities that involve a quadratic expression, such as x^2 + 4x + 4 > 0.
• Rational inequalities: These are inequalities that involve a rational expression, such as (x - 2) / (x + 1) > 0.

Q: How do I solve quadratic inequalities?

A: To solve quadratic inequalities, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.

Q: How do I solve rational inequalities?

A: To solve rational inequalities, you need to find the zeros of the numerator and denominator and then use the sign of the rational expression to determine the solution.

Conclusion

Solving inequalities is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve inequalities and write equivalent inequalities in terms of the variable. Remember to isolate the variable, check the direction of the inequality, and consider the domain of the variable to ensure that your solution is valid.