Not using the correct symbols and notation

Oct 13, 2025 by wbrum.org 101 views

Problem Description

The problem states that the sum of 11 and three-fourths of a number is less than 112. We need to write an inequality to represent this problem and find all the possible values of the number.

Writing the Inequality

Let's use the variable $x$ to represent the number. We are given that the sum of 11 and three-fourths of the number is less than 112. This can be written as:

11 + \frac{3}{4}x < 112

To isolate the $x$ term, we need to subtract 11 from both sides of the inequality:

\frac{3}{4}x < 112 - 11

Simplifying the right-hand side, we get:

\frac{3}{4}x < 101

Solving for $x$

To solve for $x$ , we need to multiply both sides of the inequality by the reciprocal of $\frac{3}{4}$ , which is $\frac{4}{3}$ . This gives us:

x < \frac{4}{3} \cdot 101

Simplifying the right-hand side, we get:

x < \frac{404}{3}

Finding the Possible Values of $x$

Since $x$ is less than $\frac{404}{3}$ , we can write the possible values of $x$ as:

x \in \left(-\infty, \frac{404}{3}\right)

This means that $x$ can take on any value less than $\frac{404}{3}$ .

Graphical Representation

We can represent the solution set graphically by drawing a number line and shading the region to the left of $\frac{404}{3}$ .

Conclusion

In this problem, we wrote an inequality to represent the given problem and found the possible values of the number. We used the variable $x$ to represent the number and isolated the $x$ term by subtracting 11 from both sides of the inequality. We then solved for $x$ by multiplying both sides of the inequality by the reciprocal of $\frac{3}{4}$ . Finally, we found the possible values of $x$ and represented the solution set graphically.

Example Use Case

Suppose we want to find the possible values of $x$ such that the sum of 11 and three-fourths of $x$ is less than 112. We can use the inequality we derived earlier:

x < \frac{404}{3}

This means that $x$ can take on any value less than $\frac{404}{3}$ .

Real-World Application

This problem has real-world applications in various fields such as finance, economics, and engineering. For example, in finance, we may want to find the possible values of a stock's price such that the sum of the initial investment and the interest earned is less than a certain threshold. In economics, we may want to find the possible values of a country's GDP such that the sum of the government's revenue and the private sector's investment is less than a certain threshold. In engineering, we may want to find the possible values of a system's output such that the sum of the input and the system's response is less than a certain threshold.

Tips and Tricks

When writing an inequality, make sure to use the correct symbols and notation.
When solving for a variable, make sure to isolate the variable term by adding or subtracting the same value from both sides of the inequality.
When finding the possible values of a variable, make sure to represent the solution set graphically by drawing a number line and shading the region to the left or right of the critical value.

Common Mistakes

When writing an inequality, make sure to use the correct symbols and notation. For example, the inequality $x > 3$ is different from the inequality $x \geq 3$ .
When solving for a variable, make sure to isolate the variable term by adding or subtracting the same value from both sides of the inequality. For example, the inequality $x + 2 < 5$ is different from the inequality $x < 5 - 2$ .
When finding the possible values of a variable, make sure to represent the solution set graphically by drawing a number line and shading the region to the left or right of the critical value. For example, the solution set for the inequality $x < 3$ is different from the solution set for the inequality $x > 3$ .

Conclusion

In conclusion, writing an inequality to represent a problem and finding the possible values of a variable is an important skill in mathematics and has real-world applications in various fields. By following the steps outlined in this article, we can write an inequality to represent a problem and find the possible values of a variable. We can also represent the solution set graphically by drawing a number line and shading the region to the left or right of the critical value.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other. Inequalities are used to describe relationships between variables and are an essential part of mathematics and problem-solving.

Q: How do I write an inequality to represent a problem?

A: To write an inequality to represent a problem, you need to identify the variable and the relationship between the variable and the other values in the problem. You can then use mathematical symbols and notation to write the inequality.

Q: What are the different types of inequalities?

A: There are four main types of inequalities:

Greater than (>): $x > a$
Less than (<): $x < a$
Greater than or equal to (≥): $x \geq a$
Less than or equal to (≤): $x \leq a$

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by adding or subtracting the same value from both sides of the inequality. You can also multiply or divide both sides of the inequality by the same value, but be careful not to change the direction of the inequality.

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 $ax + b < c$ or $ax + b > c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I graph an inequality?

A: To graph an inequality, you need to draw a number line and shade the region to the left or right of the critical value. The critical value is the value that makes 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 that satisfy the inequality. It is the set of all values that make the inequality true.

Q: How do I find the possible values of a variable in an inequality?

A: To find the possible values of a variable in an inequality, you need to solve the inequality and find the critical value. You can then use the critical value to determine the possible values of the variable.

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

A: Some common mistakes to avoid when working with inequalities include:

Not using the correct symbols and notation
Not isolating the variable
Changing the direction of the inequality
Not considering the critical value

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

A: Inequalities can be applied to a wide range of real-world problems, including finance, economics, engineering, and more. You can use inequalities to model relationships between variables and make predictions about future outcomes.

Q: What are some tips for solving inequalities?

A: Some tips for solving inequalities include:

Read the problem carefully and identify the variable and the relationship between the variable and the other values
Use the correct symbols and notation
Isolate the variable by adding or subtracting the same value from both sides of the inequality
Consider the critical value and the direction of the inequality

Q: How do I check my work when solving an inequality?

A: To check your work when solving an inequality, you need to plug in a value that satisfies the inequality and make sure that the inequality is true. You can also use a graphing calculator or a computer program to check your work.

Q: What are some common applications of inequalities?

A: Some common applications of inequalities include:

Finance: Inequalities can be used to model relationships between interest rates, investment returns, and other financial variables.
Economics: Inequalities can be used to model relationships between supply and demand, prices, and other economic variables.
Engineering: Inequalities can be used to model relationships between physical variables, such as distance, speed, and time.
Science: Inequalities can be used to model relationships between scientific variables, such as temperature, pressure, and volume.