Identify The Graph That Shows The Solutions To The Inequality 8 ( 1 + 2 X ) \textless 32 8(1+2x) \ \textless \ 32 8 ( 1 + 2 X ) \textless 32 .

Mar 13, 2025 by ADMIN 145 views

**Solving Inequalities: A Graphical Approach**

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $8(1+2x) \ \textless \ 32$ and identifying the graph that represents the solutions.

Understanding the Inequality

The given inequality is $8(1+2x) \ \textless \ 32$ . To solve this inequality, we need to isolate the variable $x$ . The first step is to distribute the coefficient $8$ to the terms inside the parentheses.

Distributing the Coefficient

Using the distributive property, we can rewrite the inequality as:

8(1+2x) = 8(1) + 8(2x)

Simplifying the expression, we get:

8 + 16x \ \textless \ 32

Isolating the Variable

The next step is to isolate the variable $x$ . To do this, we need to get rid of the constant term $8$ on the left-hand side of the inequality.

Subtracting the Constant Term

Subtracting $8$ from both sides of the inequality, we get:

16x \ \textless \ 24

Solving for x

Now that we have isolated the variable $x$ , we can solve for its value. To do this, we need to divide both sides of the inequality by the coefficient $16$ .

Dividing by the Coefficient

Dividing both sides of the inequality by $16$ , we get:

x \ \textless \ \frac{24}{16}

Simplifying the fraction, we get:

x \ \textless \ \frac{3}{2}

Graphing the Solution

The solution to the inequality $x \ \textless \ \frac{3}{2}$ can be represented graphically on a number line. The number line is a visual representation of the real numbers, and it can be used to identify the solution set.

Drawing the Number Line

To draw the number line, we need to identify the key points. The key points are the values of $x$ that satisfy the inequality. In this case, the key point is $\frac{3}{2}$ .

Shading the Region

Once we have identified the key point, we can shade the region to the left of the key point. This represents the solution set.

Conclusion

In conclusion, the graph that shows the solutions to the inequality $8(1+2x) \ \textless \ 32$ is a number line with the region to the left of $\frac{3}{2}$ shaded. This represents the solution set, which is all real numbers less than $\frac{3}{2}$ .

Key Takeaways

The inequality $8(1+2x) \ \textless \ 32$ can be solved by distributing the coefficient, isolating the variable, and solving for its value.
The solution to the inequality can be represented graphically on a number line.
The number line is a visual representation of the real numbers, and it can be used to identify the solution set.
The solution set is all real numbers less than $\frac{3}{2}$ .

Real-World Applications

Solving inequalities has many real-world applications. For example, in finance, inequalities can be used to model the growth of investments. In medicine, inequalities can be used to model the spread of diseases. In engineering, inequalities can be used to model the behavior of complex systems.

Common Mistakes

When solving inequalities, there are several common mistakes to avoid. These include:

Not distributing the coefficient correctly
Not isolating the variable correctly
Not solving for the variable correctly
Not representing the solution graphically

Tips and Tricks

When solving inequalities, there are several tips and tricks to keep in mind. These include:

Always distribute the coefficient correctly
Always isolate the variable correctly
Always solve for the variable correctly
Always represent the solution graphically

Conclusion

In conclusion, solving inequalities is an important concept in mathematics that has many real-world applications. By understanding how to solve inequalities, we can model complex systems, make informed decisions, and solve real-world problems.

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Q: What is an inequality?

A: An inequality is a statement that compares two or more expressions using a relation such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations that do not change the direction of the inequality. This may involve adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

A: A linear inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants and x is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form ax^2 + bx + c < d, where a, b, c, and d are constants and x is the variable.

Q: How do I graph an inequality?

A: To graph an inequality, you need to identify the key points that satisfy the inequality. For linear inequalities, the key point is the value of x that makes the inequality true. For quadratic inequalities, the key points are the values of x that make the inequality true.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of x that satisfy the inequality. This can be represented graphically on a number line or in interval notation.

Q: How do I represent the solution set graphically?

A: To represent the solution set graphically, you need to draw a number line and shade the region that corresponds to the solution set. For linear inequalities, the region is to the left or right of the key point. For quadratic inequalities, the region is between the key points.

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

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

Not distributing the coefficient correctly
Not isolating the variable correctly
Not solving for the variable correctly
Not representing the solution graphically

Q: What are some tips and tricks for solving inequalities?

A: Some tips and tricks for solving inequalities include:

Always distribute the coefficient correctly
Always isolate the variable correctly
Always solve for the variable correctly
Always represent the solution graphically

Q: How do I apply inequalities to real-world problems?

A: Inequalities can be applied to real-world problems in a variety of ways. For example, in finance, inequalities can be used to model the growth of investments. In medicine, inequalities can be used to model the spread of diseases. In engineering, inequalities can be used to model the behavior of complex systems.

Q: What are some examples of inequalities in real-world problems?

A: Some examples of inequalities in real-world problems include:

Modeling the growth of investments in finance
Modeling the spread of diseases in medicine
Modeling the behavior of complex systems in engineering
Modeling the cost of production in economics

Q: How do I use inequalities to make informed decisions?

A: Inequalities can be used to make informed decisions by modeling complex systems and predicting outcomes. For example, in finance, inequalities can be used to model the growth of investments and predict the return on investment. In medicine, inequalities can be used to model the spread of diseases and predict the impact of interventions.

Q: What are some common applications of inequalities in mathematics?

A: Some common applications of inequalities in mathematics include:

Modeling the growth of populations
Modeling the spread of diseases
Modeling the behavior of complex systems
Modeling the cost of production

Q: How do I use inequalities to solve problems in mathematics?

A: Inequalities can be used to solve problems in mathematics by modeling complex systems and predicting outcomes. For example, in algebra, inequalities can be used to solve systems of equations. In calculus, inequalities can be used to model the behavior of functions.

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

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

Not distributing the coefficient correctly
Not isolating the variable correctly
Not solving for the variable correctly
Not representing the solution graphically

Q: What are some tips and tricks for using inequalities to solve problems?

A: Some tips and tricks for using inequalities to solve problems include:

Always distribute the coefficient correctly
Always isolate the variable correctly
Always solve for the variable correctly
Always represent the solution graphically