Solve The Inequality:$4|x+2|-10\ \textless \ 6$

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Introduction

Inequalities are mathematical expressions that compare two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $4|x+2|-10 < 6$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Absolute Value

Before we dive into solving the inequality, let's review the concept of absolute value. The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$. The absolute value function has two cases:

• If $x \geq 0$, then $|x| = x$.
• If $x < 0$, then $|x| = -x$.

Step 1: Isolate the Absolute Value Expression

The given inequality is $4|x+2|-10 < 6$. Our first step is to isolate the absolute value expression. We can do this by adding $10$ to both sides of the inequality:

$$4|x+2| < 16$$

Step 2: Divide by 4

Next, we divide both sides of the inequality by $4$ to get:

$$|x+2| < 4$$

Step 3: Split into Two Cases

Since the absolute value expression is less than $4$, we can split the inequality into two cases:

• Case 1: $x+2 \geq 0$
• Case 2: $x+2 < 0$

Case 1: $x+2 \geq 0$

In this case, the absolute value expression simplifies to $x+2$. We can rewrite the inequality as:

$$x+2 < 4$$

Subtracting $2$ from both sides gives us:

$$x < 2$$

Case 2: $x+2 < 0$

In this case, the absolute value expression simplifies to $-(x+2)$. We can rewrite the inequality as:

$$-(x+2) < 4$$

Multiplying both sides by $-1$ gives us:

$$x+2 > -4$$

Subtracting $2$ from both sides gives us:

$$x > -6$$

Combining the Cases

We have two cases: $x < 2$ and $x > -6$. However, we need to combine these cases to get the final solution. Since the inequality is strict (less than), we need to find the values of $x$ that satisfy both cases.

The final solution is the union of the two cases:

$$x \in (-6, 2)$$

Conclusion

In this article, we solved the inequality $4|x+2|-10 < 6$ by isolating the absolute value expression, dividing by $4$, and splitting into two cases. We then combined the cases to get the final solution. The final solution is the union of the two cases, which is $x \in (-6, 2)$. This means that the values of $x$ that satisfy the inequality are all real numbers between $-6$ and $2$, excluding $-6$ and $2$.

Example Use Cases

Solving inequalities is an essential skill in mathematics, and it has many practical applications in real-life situations. Here are a few example use cases:

• Finance: In finance, inequalities are used to model financial transactions and investments. For example, a bank may use an inequality to determine the minimum amount of money that a customer needs to deposit in order to avoid a penalty.
• Science: In science, inequalities are used to model physical phenomena and make predictions. For example, a scientist may use an inequality to determine the maximum temperature that a certain material can withstand.
• Engineering: In engineering, inequalities are used to design and optimize systems. For example, an engineer may use an inequality to determine the minimum amount of energy required to power a certain device.

Tips and Tricks

Here are a few tips and tricks to help you solve inequalities:

• Use absolute value: Absolute value is a powerful tool for solving inequalities. It allows you to simplify complex inequalities and make them easier to solve.
• Split into cases: When solving inequalities, it's often helpful to split the inequality into two or more cases. This can help you identify the values of the variable that satisfy the inequality.
• Combine cases: Once you've split the inequality into cases, you need to combine them to get the final solution. This can be a bit tricky, but it's an essential step in solving inequalities.

Conclusion

Introduction

In our previous article, we solved the inequality $4|x+2|-10 < 6$ by isolating the absolute value expression, dividing by $4$, and splitting into two cases. We then combined the cases to get the final solution. In this article, we will answer some frequently asked questions about solving inequalities.

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

A: An inequality is a mathematical expression that compares two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to. An equation, on the other hand, is a mathematical expression that states that two values are equal.

Q: How do I know which inequality symbol to use?

A: The inequality symbol you use depends on the problem you are trying to solve. For example, if you are trying to find the values of $x$ that are greater than $5$, you would use the inequality symbol $x > 5$. If you are trying to find the values of $x$ that are less than $5$, you would use the inequality symbol $x < 5$.

Q: What is the absolute value function?

A: The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$.

Q: How do I isolate the absolute value expression in an inequality?

A: To isolate the absolute value expression in an inequality, you need to get rid of any constants or variables that are being added or subtracted from the absolute value expression. You can do this by adding or subtracting the same value from both sides of the inequality.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality symbol, such as $x < 5$ or $x > 5$. A non-strict inequality is an inequality that is written with a non-strict inequality symbol, such as $x \leq 5$ or $x \geq 5$.

Q: How do I solve an inequality with a variable in the denominator?

A: To solve an inequality with a variable in the denominator, you need to get rid of the variable in the denominator by multiplying both sides of the inequality by the reciprocal of the denominator.

Q: What is the final solution to an inequality?

A: The final solution to an inequality is the set of values that satisfy the inequality. This can be a single value, a range of values, or a combination of both.

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, and then draw an arrow on the number line that represents the direction of 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 absolute value expression
• Not splitting the inequality into cases
• Not combining the cases correctly
• Not checking the solution for extraneous solutions

Conclusion

Solving inequalities is an essential skill in mathematics, and it has many practical applications in real-life situations. By following the steps outlined in this article, you can solve inequalities with ease and confidence. Remember to use absolute value, split into cases, and combine cases to get the final solution. With practice and patience, you'll become a master of solving inequalities in no time!

Example Problems

Here are a few example problems to help you practice solving inequalities:

• Solve the inequality $3|x-2|+5 < 10$.
• Solve the inequality $2|x+1|-3 > 5$.
• Solve the inequality $x^2+4x-5 > 0$.

Tips and Tricks

Here are a few tips and tricks to help you solve inequalities:

• Use absolute value to simplify complex inequalities.
• Split the inequality into cases to make it easier to solve.
• Combine the cases correctly to get the final solution.
• Check the solution for extraneous solutions.

Conclusion

Solving inequalities is an essential skill in mathematics, and it has many practical applications in real-life situations. By following the steps outlined in this article, you can solve inequalities with ease and confidence. Remember to use absolute value, split into cases, and combine cases to get the final solution. With practice and patience, you'll become a master of solving inequalities in no time!