Solve The Inequality And Then Graph The Solution On The Number Line.${ -3(4x - 2) \leq 14x - 3 }$

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 and graphing inequalities is an essential skill in mathematics, particularly in algebra and calculus. In this article, we will focus on solving and graphing the inequality ${-3(4x - 2) \leq 14x - 3}$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Distribute and Simplify the Inequality

The first step in solving the inequality is to distribute the negative 3 to the terms inside the parentheses.

${ -3(4x - 2) \leq 14x - 3 }$

Using the distributive property, we get:

${ -12x + 6 \leq 14x - 3 }$

Next, we will simplify the inequality by combining like terms.

Step 2: Combine Like Terms

To combine like terms, we need to move all the x terms to one side of the inequality and the constants to the other side.

${ -12x + 6 \leq 14x - 3 }$

Subtracting 6 from both sides gives us:

${ -12x \leq 14x - 9 }$

Adding 12x to both sides gives us:

${ 0 \leq 26x - 9 }$

Step 3: Isolate the Variable

Now, we need to isolate the variable x by adding 9 to both sides of the inequality.

${ 0 \leq 26x - 9 }$

Adding 9 to both sides gives us:

${ 9 \leq 26x }$

Dividing both sides by 26 gives us:

${ \frac{9}{26} \leq x }$

Step 4: Graph the Solution on the Number Line

To graph the solution on the number line, we need to find the boundary point of the inequality. The boundary point is the value of x that makes the inequality true.

In this case, the boundary point is ${\frac{9}{26}}$. To graph the solution, we will draw a closed circle at the boundary point and shade the region to the right of the boundary point.

Graphing the Solution

Here is the graph of the solution:

• The boundary point is ${\frac{9}{26}}$.
• The solution is the region to the right of the boundary point.
• The graph is a closed circle at the boundary point.

Conclusion

Solving and graphing inequalities is an essential skill in mathematics. By following the steps outlined in this article, you can solve and graph the inequality ${-3(4x - 2) \leq 14x - 3}$. Remember to distribute and simplify the inequality, combine like terms, isolate the variable, and graph the solution on the number line.

Tips and Tricks

• Always check your work by plugging in values into the inequality.
• Use a number line to graph the solution.
• Make sure to label the boundary point and the solution region.

Common Mistakes

• Forgetting to distribute and simplify the inequality.
• Not combining like terms.
• Not isolating the variable.
• Not graphing the solution on the number line.

Real-World Applications

Solving and graphing inequalities has many real-world applications, including:

• Optimization problems
• Data analysis
• Business decision-making
• Science and engineering

Introduction

In our previous article, we discussed how to solve and graph the inequality ${-3(4x - 2) \leq 14x - 3}$. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving and graphing inequalities.

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: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities involve a linear expression, while quadratic inequalities involve a quadratic expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow these steps:

1. Distribute and simplify the inequality.
2. Combine like terms.
3. Isolate the variable.
4. Graph the solution on the number line.

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

A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression. Quadratic inequalities are more complex and require additional techniques to solve.

Q: How do I graph a solution on the number line?

A: To graph a solution on the number line, you need to follow these steps:

1. Find the boundary point of the inequality.
2. Draw a closed circle at the boundary point.
3. Shade the region to the right of the boundary point (for greater than or equal to inequalities) or to the left of the boundary point (for less than or equal to inequalities).

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

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

• Forgetting to distribute and simplify the inequality.
• Not combining like terms.
• Not isolating the variable.
• Not graphing the solution on the number line.

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

A: Solving and graphing inequalities has many real-world applications, including:

• Optimization problems
• Data analysis
• Business decision-making
• Science and engineering

Q: How can I practice solving and graphing inequalities?

A: You can practice solving and graphing inequalities by:

• Working on sample problems.
• Using online resources and tutorials.
• Practicing with real-world applications.

Conclusion

Solving and graphing inequalities is an essential skill in mathematics. By following the steps outlined in this article and practicing with sample problems, you can master the skills involved in solving and graphing inequalities.

Tips and Tricks

• Always check your work by plugging in values into the inequality.
• Use a number line to graph the solution.
• Make sure to label the boundary point and the solution region.

Common Mistakes

• Forgetting to distribute and simplify the inequality.
• Not combining like terms.
• Not isolating the variable.
• Not graphing the solution on the number line.

Real-World Applications

Solving and graphing inequalities has many real-world applications, including:

• Optimization problems
• Data analysis
• Business decision-making
• Science and engineering

By mastering the skills outlined in this article, you can apply them to real-world problems and make informed decisions.