Solve The Inequality: − 5 ( X − 1 ) \textgreater − 40 -5(x - 1) \ \textgreater \ -40 − 5 ( X − 1 ) \textgreater − 40

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 focus on solving the inequality $-5(x - 1) > -40$.

Understanding the Inequality

The given inequality is $-5(x - 1) > -40$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by distributing the negative 5 to the terms inside the parentheses.

Distributing the Negative 5

When we distribute the negative 5 to the terms inside the parentheses, we get:

$$-5(x - 1) = -5x + 5$$

So, the inequality becomes:

$$-5x + 5 > -40$$

Isolating the Variable

The next step is to isolate the variable $x$ on one side of the inequality sign. To do this, we need to get rid of the constant term 5 on the left-hand side of the inequality. We can do this by subtracting 5 from both sides of the inequality.

$$-5x + 5 - 5 > -40 - 5$$

This simplifies to:

$$-5x > -45$$

Dividing Both Sides

The next step is to divide both sides of the inequality by -5. However, when we divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

$$\frac{-5x}{-5} > \frac{-45}{-5}$$

This simplifies to:

$$x < 9$$

Conclusion

In this article, we solved the inequality $-5(x - 1) > -40$ by isolating the variable $x$ on one side of the inequality sign. We simplified the left-hand side of the inequality by distributing the negative 5 to the terms inside the parentheses, and then isolated the variable by subtracting 5 from both sides of the inequality. Finally, we divided both sides of the inequality by -5 and reversed the direction of the inequality sign.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always simplify the left-hand side of the inequality by distributing any negative numbers to the terms inside the parentheses.
• Isolate the variable on one side of the inequality sign by adding or subtracting the same value to both sides of the inequality.
• When dividing both sides of an inequality by a negative number, reverse the direction of the inequality sign.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to simplify the left-hand side of the inequality by distributing negative numbers.
• Not isolating the variable on one side of the inequality sign.
• Failing to reverse the direction of the inequality sign when dividing both sides of the inequality by a negative number.

Real-World Applications

Inequalities have many real-world applications. Here are a few examples:

• In finance, inequalities are used to calculate interest rates and investment returns.
• In engineering, inequalities are used to design and optimize systems.
• In medicine, inequalities are used to model and analyze the spread of diseases.

Conclusion

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Introduction

In our previous article, we solved the inequality $-5(x - 1) > -40$ by isolating the variable $x$ on one side of the inequality sign. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, while solving an inequality involves finding the values of the variable that make the inequality true.

Q: How do I know which direction to flip the inequality sign when dividing both sides by a negative number?

A: When dividing both sides of an inequality by a negative number, you need to flip the inequality sign. For example, if you have the inequality $x > 5$ and you divide both sides by -2, the inequality becomes $x < -\frac{5}{2}$.

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. For example, if you have the inequality $x > 5$ and you add 3 to both sides, the inequality becomes $x + 3 > 8$.

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 plug in a value for the variable and see if the inequality is true or false. For example, if you have the inequality $x > 5$ and you plug in the value $x = 6$, the inequality is true.

Q: Can I multiply or divide both sides of an inequality by a fraction?

A: Yes, you can multiply or divide both sides of an inequality by a fraction. However, you need to be careful when multiplying or dividing by a fraction that is not equal to 1. For example, if you have the inequality $x > 5$ and you multiply both sides by $\frac{1}{2}$, the inequality becomes $x > \frac{5}{2}$.

Q: How do I solve a compound inequality?

A: A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. For example, if you have the compound inequality $x > 5$ and $x < 10$, the solution is $5 < x < 10$.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: Yes, you can use the same steps to solve a linear inequality as you would to solve a linear equation. However, you need to be careful when dividing both sides of an inequality by a negative number and when multiplying or dividing both sides of an inequality by a fraction.

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 solution to the inequality. For example, if you have the inequality $x > 5$, you would plot a point on the number line at $x = 6$ and shade the region to the right of the point.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We discussed the difference between solving an equation and solving an inequality, how to flip the inequality sign when dividing both sides by a negative number, and how to solve a compound inequality. We also discussed how to graph an inequality on a number line.