What Is The First Step In Solving $-5x - \frac{1}{6} \geq \frac{5}{2}$?A. Add $\frac{1}{6}$ To Both Sides.B. Subtract $\frac{1}{6}$ From Both Sides.C. Divide Both Sides By -5.D. Multiply Both Sides By -5.

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Understanding the Inequality

When solving an inequality, it's essential to follow the correct steps to isolate the variable. In this case, we have the inequality $-5x - \frac{1}{6} \geq \frac{5}{2}$. Our goal is to find the first step in solving this inequality.

Isolating the Variable

To isolate the variable $x$, we need to get rid of the constant term $-\frac{1}{6}$ on the left-hand side of the inequality. However, we cannot simply add or subtract the constant term from both sides because it's not isolated on one side of the inequality.

Adding $\frac{1}{6}$ to Both Sides

Let's consider option A: adding $\frac{1}{6}$ to both sides of the inequality. This would result in:

$$-5x \geq \frac{5}{2} + \frac{1}{6}$$

However, this is not the correct step because we are not isolating the variable $x$ yet. We are still dealing with the constant term $-\frac{1}{6}$ on the left-hand side.

Subtracting $\frac{1}{6}$ from Both Sides

Now, let's consider option B: subtracting $\frac{1}{6}$ from both sides of the inequality. This would result in:

$$-5x - \frac{1}{6} - \frac{1}{6} \geq \frac{5}{2} - \frac{1}{6}$$

Simplifying the left-hand side, we get:

$$-5x - \frac{1}{3} \geq \frac{5}{2} - \frac{1}{6}$$

However, this is still not the correct step because we are not isolating the variable $x$ yet.

Dividing Both Sides by -5

Now, let's consider option C: dividing both sides of the inequality by $-5$. This would result in:

$$x \leq -\frac{1}{6} + \frac{5}{10}$$

Simplifying the right-hand side, we get:

$$x \leq -\frac{1}{6} + \frac{1}{2}$$

However, this is still not the correct step because we are not isolating the variable $x$ yet.

Multiplying Both Sides by -5

Finally, let's consider option D: multiplying both sides of the inequality by $-5$. This would result in:

$$-5x \leq -\frac{1}{6} + \frac{5}{2}$$

However, this is not the correct step because we are not isolating the variable $x$ yet.

The Correct Step

The correct step in solving the inequality $-5x - \frac{1}{6} \geq \frac{5}{2}$ is to add $\frac{1}{6}$ to both sides of the inequality. This would result in:

$$-5x \geq \frac{5}{2} + \frac{1}{6}$$

However, we need to simplify the right-hand side of the inequality.

Simplifying the Right-Hand Side

To simplify the right-hand side of the inequality, we need to find a common denominator. The least common multiple of $2$ and $6$ is $6$. Therefore, we can rewrite the right-hand side as:

$$-5x \geq \frac{15}{6} + \frac{1}{6}$$

Simplifying the right-hand side, we get:

$$-5x \geq \frac{16}{6}$$

Dividing Both Sides by -5

Now that we have simplified the right-hand side of the inequality, we can divide both sides by $-5$. However, we need to be careful when dividing by a negative number. When we divide by a negative number, the direction of the inequality sign changes.

$$x \leq -\frac{16}{30}$$

Simplifying the right-hand side, we get:

$$x \leq -\frac{8}{15}$$

Conclusion

In conclusion, the first step in solving the inequality $-5x - \frac{1}{6} \geq \frac{5}{2}$ is to add $\frac{1}{6}$ to both sides of the inequality. This would result in:

$$-5x \geq \frac{5}{2} + \frac{1}{6}$$

However, we need to simplify the right-hand side of the inequality. After simplifying the right-hand side, we can divide both sides by $-5$. When we divide by a negative number, the direction of the inequality sign changes.

$$x \leq -\frac{8}{15}$$

Therefore, the correct answer is option A: add $\frac{1}{6}$ to both sides of the inequality.

Final Answer

The final answer is A.

Frequently Asked Questions

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality. This can be done by adding or subtracting the same value from both sides of the inequality.

Q: How do I add or subtract the same value from both sides of an inequality?

A: To add or subtract the same value from both sides of an inequality, you need to make sure that the value is on the same side of the inequality as the variable. For example, if you want to add 3 to both sides of the inequality x - 2 ≥ 5, you would add 2 to both sides of the inequality to get x ≥ 7.

Q: What is the difference between adding and subtracting the same value from both sides of an inequality?

A: Adding the same value to both sides of an inequality will not change the direction of the inequality sign. However, subtracting the same value from both sides of an inequality will change the direction of the inequality sign.

Q: How do I divide both sides of an inequality by a negative number?

A: When dividing 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 want to divide both sides by -2, you would get x ≤ -2.5.

Q: What is the correct order of operations when solving an inequality?

A: The correct order of operations when solving an inequality is:

1. Simplify the inequality by combining like terms.
2. Add or subtract the same value from both sides of the inequality to isolate the variable.
3. Multiply or divide both sides of the inequality by a positive number to isolate the variable.
4. Change the direction of the inequality sign when dividing both sides of the inequality by a negative number.

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 that satisfies the inequality and make sure that the inequality is true. For example, if you have the inequality x ≥ 5 and you plug in x = 10, you would get 10 ≥ 5, which is true.

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

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

• Not changing the direction of the inequality sign when dividing both sides of the inequality by a negative number.
• Not simplifying the inequality by combining like terms.
• Not adding or subtracting the same value from both sides of the inequality to isolate the variable.
• Not checking your solution to the inequality.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to draw a line that represents the inequality and shade in the region that satisfies the inequality. For example, if you have the inequality x ≥ 5, you would draw a line at x = 5 and shade in the region to the right of the line.

Q: What are some real-world applications of solving inequalities?

A: Some real-world applications of solving inequalities include:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities.

Q: How do I use technology to solve inequalities?

A: There are many software programs and online tools that can be used to solve inequalities, including graphing calculators, computer algebra systems, and online inequality solvers. These tools can be used to graph inequalities, solve systems of inequalities, and check solutions to inequalities.

Q: What are some tips for solving inequalities?

A: Some tips for solving inequalities include:

• Read the problem carefully and make sure you understand what is being asked.
• Simplify the inequality by combining like terms.
• Add or subtract the same value from both sides of the inequality to isolate the variable.
• Multiply or divide both sides of the inequality by a positive number to isolate the variable.
• Change the direction of the inequality sign when dividing both sides of the inequality by a negative number.
• Check your solution to the inequality.

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

A: Inequalities can be used in a variety of real-world situations, including:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: inequalities that can be written in the form ax + b ≥ c or ax + b ≤ c.
• Quadratic inequalities: inequalities that can be written in the form ax^2 + bx + c ≥ d or ax^2 + bx + c ≤ d.
• Rational inequalities: inequalities that can be written in the form ax/b ≥ c or ax/b ≤ c.
• Absolute value inequalities: inequalities that can be written in the form |x| ≥ c or |x| ≤ c.

Q: How do I use inequalities to solve systems of equations?

A: Inequalities can be used to solve systems of equations by finding the intersection of the solution sets of the individual inequalities. This can be done by graphing the inequalities on a number line and finding the region where the two solution sets intersect.

Q: What are some common mistakes to avoid when using inequalities to solve systems of equations?

A: Some common mistakes to avoid when using inequalities to solve systems of equations include:

• Not graphing the inequalities on a number line.
• Not finding the intersection of the solution sets of the individual inequalities.
• Not checking the solution to the system of equations.

Q: How do I use inequalities to model real-world situations?

A: Inequalities can be used to model a variety of real-world situations, including:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities.

Q: What are some common types of real-world situations that can be modeled using inequalities?

A: Some common types of real-world situations that can be modeled using inequalities include:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities.

Q: How do I use inequalities to solve optimization problems?

A: Inequalities can be used to solve optimization problems by finding the maximum or minimum value of a function. This can be done by graphing the function on a number line and finding the region where the function is maximized or minimized.

Q: What are some common mistakes to avoid when using inequalities to solve optimization problems?

A: Some common mistakes to avoid when using inequalities to solve optimization problems include:

• Not graphing the function on a number line.
• Not finding the region where the function is maximized or minimized.
• Not checking the solution to the optimization problem.

Q: How do I use inequalities to model real-world situations using graphs?

A: Inequalities can be used to model real-world situations using graphs by graphing the inequalities on a number line and finding the region where the solution set is located.

Q: What are some common types of real-world situations that can be modeled using inequalities and graphs?

A: Some common types of real-world situations that can be modeled using inequalities and graphs include:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities and graphs.

Q: How do I use inequalities to solve systems of inequalities?

A: Inequalities can be used to solve systems of inequalities by finding the intersection of the solution sets of the individual inequalities. This can be done by graphing the inequalities on a number line and finding the region where the two solution sets intersect.

Q: What are some common mistakes to avoid when using inequalities to solve systems of inequalities?

A: Some common mistakes to avoid when using inequalities to solve systems of inequalities include:

• Not graphing the inequalities on a number line.
• Not finding the intersection of the solution sets of the individual inequalities.
• Not checking the solution to the system of inequalities.

Q: How do I use inequalities to model real-world situations using systems of inequalities?

A: Inequalities can be used to model real-world situations using systems of inequalities by finding the intersection of the solution sets of the individual inequalities. This can be done by graphing the inequalities on a number line and finding the region where the two solution sets intersect.

Q: What are some common types of real-world situations that can be modeled using inequalities and systems of inequalities?

A: Some common types of real-world situations that can be modeled using inequalities and systems of inequalities include:

• Finding the maximum or minimum value of a function.
• Determining the range of values for a variable.
• Solving optimization problems.
• Modeling real-world situations using inequalities and systems of inequalities.

Q: How do I use inequalities to solve optimization problems using systems of inequalities?

A: Inequalities can be used to solve optimization problems using systems of inequalities by finding the maximum or minimum value of a function. This can be done by graphing the function on a number line and finding the region where the function is maximized or minimized.

Q: What are some common mistakes to avoid when using inequalities to solve optimization problems using systems of inequalities?

A: Some common mistakes to avoid when using inequalities to solve optimization problems using systems of inequalities include:

• Not graphing the function on a number line.
• Not finding the region where the function is maximized or minimized.
• Not checking the solution to the optimization