Solve The Inequality:1. { -4 \ \textless \ K + 3 \ \textless \ 8$}$Graph Your Solution.

Mar 10, 2025 by ADMIN 92 views

**Solving Inequalities: A Step-by-Step Guide**

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and solve problems. In this article, we will focus on solving a specific inequality and graphing the solution. We will use the given inequality: $-4 < k + 3 < 8$ . Our goal is to isolate the variable $k$ and find the range of values that satisfy the inequality.

Understanding the Inequality

Before we start solving the inequality, let's understand what it means. The inequality $-4 < k + 3 < 8$ can be read as: " $k + 3$ is greater than $-4$ and less than $8$ ". This means that the value of $k + 3$ must be between $-4$ and $8$ , but not equal to either of these values.

Step 1: Subtract 3 from all parts of the inequality

To isolate the variable $k$ , we need to get rid of the constant term $3$ that is being added to $k$ . We can do this by subtracting $3$ from all parts of the inequality. This gives us:

-4 - 3 < k + 3 - 3 < 8 - 3

Simplifying the inequality, we get:

-7 < k < 5

Step 2: Write the solution in interval notation

The solution to the inequality $-7 < k < 5$ can be written in interval notation as $(-7, 5)$ . This means that the value of $k$ must be greater than $-7$ and less than $5$ , but not equal to either of these values.

Graphing the Solution

To graph the solution, we can use a number line. We start by marking the point $-7$ on the number line, and then draw an open circle around it to indicate that $k$ is not equal to $-7$ . We then mark the point $5$ on the number line, and draw an open circle around it to indicate that $k$ is not equal to $5$ . Finally, we draw a line segment between the two points to indicate that $k$ is greater than $-7$ and less than $5$ .

Conclusion

In this article, we solved the inequality $-4 < k + 3 < 8$ and graphed the solution. We used the steps of subtracting $3$ from all parts of the inequality and writing the solution in interval notation. We also graphed the solution using a number line. The solution to the inequality is $(-7, 5)$ , which means that the value of $k$ must be greater than $-7$ and less than $5$ , but not equal to either of these values.

Tips and Tricks

When solving inequalities, it's essential to remember that the inequality sign can be flipped when multiplying or dividing both sides of the inequality by a negative number.
When graphing the solution, use a number line to visualize the range of values that satisfy the inequality.
When writing the solution in interval notation, use parentheses to indicate that the value is not included in the interval.

Common Mistakes

When solving inequalities, it's easy to get confused and flip the inequality sign incorrectly. Make sure to double-check your work and use the correct inequality sign.
When graphing the solution, it's easy to forget to include the endpoints of the interval. Make sure to include the endpoints and use open circles to indicate that the value is not included in the interval.

Real-World Applications

Inequalities are used in a wide range of real-world applications, including:

Finance: Inequalities are used to calculate interest rates and investment returns.
Science: Inequalities are used to model population growth and decay.
Engineering: Inequalities are used to design and optimize systems.

Practice Problems

Solve the inequality $2x - 5 < 3x + 2 < 11$ and graph the solution.
Solve the inequality $-3 < 2y - 1 < 7$ and graph the solution.
Solve the inequality $4x + 2 > 2x - 3 > 5$ and graph the solution.

Conclusion

Introduction

In our previous article, we solved the inequality $-4 < k + 3 < 8$ and graphed the solution. In this article, we will answer some common questions that students often have when solving inequalities.

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

A: An inequality is a statement that compares two values using a symbol such as <, >, ≤, or ≥. An equation is a statement that says two values are equal. For example, $x + 3 = 5$ is an equation, while $x + 3 < 5$ is an inequality.

Q: How do I solve an inequality with a variable on both sides?

A: To solve an inequality with a variable on both sides, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality. For example, to solve the inequality $x + 2 = 3x - 1$ , you would add $-x$ to both sides to get $2 = 2x - 1$ , and then add $1$ to both sides to get $3 = 2x$ . Finally, you would divide both sides by $2$ to get $x = \frac{3}{2}$ .

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

A: A strict inequality is an inequality that uses a symbol such as < or >, while a non-strict inequality is an inequality that uses a symbol such as ≤ or ≥. For example, $x < 5$ is a strict inequality, while $x ≤ 5$ is a non-strict inequality.

Q: How do I graph the solution to an inequality?

A: To graph the solution to an inequality, you need to use a number line. You start by marking the point that is equal to the value on the right-hand side of the inequality, and then draw an open circle around it to indicate that the value is not included in the solution. You then draw a line segment to the left or right of the point, depending on whether the inequality is strict or non-strict.

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 solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the expression to determine the solution. For example, to solve the inequality $x^2 + 4x + 4 > 0$ , you would factor the expression as $(x + 2)^2 > 0$ , and then use the sign of the expression to determine that the solution is $x < -2$ or $x > -2$ .

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} < 0$ or $\frac{f(x)}{g(x)} > 0$ , where $f(x)$ and $g(x)$ are polynomials. A polynomial inequality is an inequality that can be written in the form $f(x) < 0$ or $f(x) > 0$ , where $f(x)$ is a polynomial.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the zeros of the numerator and denominator, and then use the sign of the expression to determine the solution. For example, to solve the inequality $\frac{x - 1}{x + 1} < 0$ , you would find the zeros of the numerator and denominator as $x = 1$ and $x = -1$ , and then use the sign of the expression to determine that the solution is $x < -1$ or $x > 1$ .

Conclusion

In this article, we answered some common questions that students often have when solving inequalities. We covered topics such as the difference between an inequality and an equation, how to solve an inequality with a variable on both sides, and how to graph the solution to an inequality. We also covered more advanced topics such as quadratic inequalities and rational inequalities. We hope that this article has been helpful in answering your questions and providing you with a better understanding of inequalities.