Solve The Inequality:$ 3(x - 2) \leq 19 - 2x $

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Introduction

In this article, we will focus on solving the given inequality $3(x - 2) \leq 19 - 2x$. This involves isolating the variable $x$ and determining the range of values that satisfy the inequality. We will use algebraic manipulations and properties of inequalities to solve the problem.

Understanding the Inequality

The given inequality is $3(x - 2) \leq 19 - 2x$. To begin solving this inequality, we need to understand the properties of inequalities. Inequalities are mathematical statements that compare two expressions, indicating whether one is greater than, less than, or equal to the other. In this case, we have a less-than-or-equal-to inequality.

Distributing and Simplifying

To solve the inequality, we can start by distributing the 3 to the terms inside the parentheses: $3x - 6 \leq 19 - 2x$. This simplifies the left-hand side of the inequality.

Combining Like Terms

Next, we can combine like terms on both sides of the inequality. On the left-hand side, we have $3x$ and $-6$. On the right-hand side, we have $19$ and $-2x$. Combining like terms, we get $3x - 2x \leq 19 + 6$.

Isolating the Variable

Now, we can isolate the variable $x$ by combining the like terms on the left-hand side: $x \leq 25$. This is the solution to the inequality.

Checking the Solution

To verify the solution, we can plug in a value of $x$ that satisfies the inequality and check if the inequality holds true. Let's choose $x = 20$. Plugging this value into the original inequality, we get $3(20 - 2) \leq 19 - 2(20)$, which simplifies to $54 \leq -21$. This is not true, so the solution $x \leq 25$ is not valid.

Revising the Solution

Since the solution $x \leq 25$ is not valid, we need to revise it. Let's go back to the previous step and re-examine the inequality. We had $3x - 2x \leq 19 + 6$, which simplifies to $x \leq 25$. However, we made an error in our previous step. Let's correct it.

Correcting the Error

The correct simplification of the inequality is $x + 6 \leq 25$. Subtracting 6 from both sides, we get $x \leq 19$. This is the correct solution to the inequality.

Conclusion

In conclusion, the solution to the inequality $3(x - 2) \leq 19 - 2x$ is $x \leq 19$. This involves isolating the variable $x$ and determining the range of values that satisfy the inequality. We used algebraic manipulations and properties of inequalities to solve the problem.

Final Answer

The final answer is $\boxed{x \leq 19}$.

Additional Tips and Tricks

• When solving inequalities, it's essential to keep track of the direction of the inequality.
• Use algebraic manipulations and properties of inequalities to simplify the inequality.
• Check the solution by plugging in a value of $x$ that satisfies the inequality.

Common Mistakes to Avoid

• Don't forget to distribute the terms inside the parentheses.
• Be careful when combining like terms.
• Make sure to check the solution by plugging in a value of $x$ that satisfies the inequality.

Real-World Applications

Inequalities have numerous real-world applications, such as:

• Modeling population growth
• Determining the maximum or minimum value of a function
• Finding the range of values that satisfy a given condition

Final Thoughts

Solving inequalities requires a deep understanding of algebraic manipulations and properties of inequalities. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to keep track of the direction of the inequality, use algebraic manipulations and properties of inequalities to simplify the inequality, and check the solution by plugging in a value of $x$ that satisfies the inequality.

Introduction

In the previous article, we discussed how to solve the inequality $3(x - 2) \leq 19 - 2x$. In this article, we will provide a Q&A section to help you better understand the concepts and techniques involved in solving inequalities.

Q1: What is the first step in solving an inequality?

A1: The first step in solving an inequality is to simplify the inequality by combining like terms and using algebraic manipulations.

Q2: How do I determine the direction of the inequality?

A2: To determine the direction of the inequality, you need to consider the signs of the coefficients of the variable. If the coefficient is positive, the inequality is "greater than or equal to" or "less than or equal to". If the coefficient is negative, the inequality is "less than or greater than".

Q3: What is the difference between a linear inequality and a quadratic inequality?

A3: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q4: How do I solve a quadratic inequality?

A4: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.

Q5: What is the importance of checking the solution?

A5: Checking the solution is essential to ensure that the solution set is correct. If the solution set is not correct, it can lead to incorrect conclusions and decisions.

Q6: How do I graph an inequality?

A6: To graph an inequality, you need to plot the boundary line and then test points in each region to determine which region satisfies the inequality.

Q7: What is the difference between a linear inequality and a nonlinear inequality?

A7: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A nonlinear inequality is an inequality that cannot be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants.

Q8: How do I solve a nonlinear inequality?

A8: To solve a nonlinear inequality, you need to use algebraic manipulations and properties of inequalities to simplify the inequality and then use the sign of the nonlinear expression to determine the solution set.

Q9: What is the importance of understanding the concept of absolute value?

A9: Understanding the concept of absolute value is essential in solving inequalities, as it allows you to simplify the inequality and determine the solution set.

Q10: How do I use the concept of absolute value to solve an inequality?

A10: To use the concept of absolute value to solve an inequality, you need to rewrite the inequality in terms of absolute value and then use the properties of absolute value to simplify the inequality and determine the solution set.

Conclusion

In conclusion, solving inequalities requires a deep understanding of algebraic manipulations and properties of inequalities. By following the steps outlined in this article and using the Q&A section, you can better understand the concepts and techniques involved in solving inequalities.

Final Thoughts

Solving inequalities is an essential skill in mathematics, and it has numerous real-world applications. By mastering the concepts and techniques involved in solving inequalities, you can solve problems with confidence and make informed decisions.

Additional Resources

• For more information on solving inequalities, please refer to the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Common Mistakes to Avoid

• Don't forget to simplify the inequality by combining like terms and using algebraic manipulations.
• Be careful when determining the direction of the inequality.
• Make sure to check the solution by plugging in a value of $x$ that satisfies the inequality.

Real-World Applications

Inequalities have numerous real-world applications, such as:

• Modeling population growth
• Determining the maximum or minimum value of a function
• Finding the range of values that satisfy a given condition

Final Answer

The final answer is $\boxed{0}$.