Solve The Inequality, Then Identify The Graph Of The Solution.${ -3x - 3 \leq 6 }$

Mar 10, 2025 by ADMIN 84 views

Solving Inequalities and Graphing Solutions

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Introduction

In mathematics, inequalities are used to compare two or more values. They are an essential part of algebra and are used to solve a wide range of problems. In this article, we will focus on solving the inequality $-3x - 3 \leq 6$ and then identify the graph of the solution.

Understanding Inequalities

Inequalities are mathematical statements that compare two or more values using greater than, less than, greater than or equal to, or less than or equal to. They can be written in the form of an equation, but with a different symbol. The most common inequality symbols are:

Greater than: $>$
Less than: $<$
Greater than or equal to: $\geq$
Less than or equal to: $\leq$

Solving the Inequality

To solve the inequality $-3x - 3 \leq 6$ , we need to isolate the variable $x$ . We can do this by adding $3$ to both sides of the inequality.

-3x - 3 + 3 \leq 6 + 3

This simplifies to:

-3x \leq 9

Next, we need to get rid of the negative sign in front of the $x$ . We can do this by dividing both sides of the inequality by $-3$ . However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality.

\frac{-3x}{-3} \geq \frac{9}{-3}

This simplifies to:

x \geq -3

Graphing the Solution

The solution to the inequality $x \geq -3$ is a range of values that $x$ can take. To graph this solution, we need to draw a number line and mark the value $-3$ with an open circle.

We then need to shade the region to the right of the value $-3$ to indicate that $x$ can take any value greater than or equal to $-3$ .

Conclusion

In this article, we solved the inequality $-3x - 3 \leq 6$ and identified the graph of the solution. We learned how to isolate the variable $x$ and how to graph the solution on a number line. Inequalities are an essential part of mathematics, and solving them is a crucial skill that can be applied to a wide range of problems.

Tips and Tricks

When solving an inequality, always isolate the variable on one side of the inequality.
When dividing or multiplying an inequality by a negative number, reverse the direction of the inequality.
When graphing the solution to an inequality, use a number line and shade the region that satisfies the inequality.

Real-World Applications

Inequalities have many real-world applications. For example:

In finance, inequalities are used to calculate interest rates and investment returns.
In science, inequalities are used to model population growth and decay.
In engineering, inequalities are used to design and optimize systems.

Practice Problems

Here are some practice problems to help you practice solving inequalities and graphing solutions:

Solve the inequality $2x + 5 \leq 11$ and graph the solution.
Solve the inequality $x - 3 \geq 7$ and graph the solution.
Solve the inequality $-4x + 2 \leq 10$ and graph the solution.

Conclusion

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Introduction

In our previous article, we solved the inequality $-3x - 3 \leq 6$ and identified the graph of the solution. In this article, we will answer some frequently asked questions about solving inequalities and graphing solutions.

Q&A

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

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value from both sides of the inequality. For example, to solve the inequality $-3x - 3 \leq 6$ , you can add $3$ to both sides of the inequality to get $-3x \leq 9$ .

Q: How do I graph the solution to a linear inequality?

A: To graph the solution to a linear inequality, you need to draw a number line and mark the value that satisfies the inequality. You then need to shade the region to the right or left of the value to indicate that the solution is greater than or less than the value.

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

A: A strict inequality is an inequality that is written with a strict symbol, such as $>$ or $<$ . A non-strict inequality is an inequality that is written with a non-strict symbol, such as $\geq$ or $\leq$ .

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 factored form to determine the solution. For example, to solve the inequality $x^2 + 4x + 4 \leq 0$ , you can factor the quadratic expression as $(x + 2)^2 \leq 0$ .

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

A: A rational inequality is an inequality that involves a rational expression, such as $\frac{x}{x+1} \leq 2$ . A polynomial inequality is an inequality that involves a polynomial expression, such as $x^2 + 4x + 4 \leq 0$ .

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the values of the variable that make the numerator and denominator equal to zero. You then need to use these values to determine the solution.

Tips and Tricks

When solving an inequality, always isolate the variable on one side of the inequality.
When dividing or multiplying an inequality by a negative number, reverse the direction of the inequality.
When graphing the solution to an inequality, use a number line and shade the region that satisfies the inequality.

Real-World Applications

Inequalities have many real-world applications. For example:

In finance, inequalities are used to calculate interest rates and investment returns.
In science, inequalities are used to model population growth and decay.
In engineering, inequalities are used to design and optimize systems.

Practice Problems

Here are some practice problems to help you practice solving inequalities and graphing solutions:

Solve the inequality $2x + 5 \leq 11$ and graph the solution.
Solve the inequality $x - 3 \geq 7$ and graph the solution.
Solve the inequality $-4x + 2 \leq 10$ and graph the solution.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities and graphing solutions. We learned how to solve linear and quadratic inequalities, how to graph the solution to an inequality, and how to solve rational and polynomial inequalities. Inequalities are an essential part of mathematics, and solving them is a crucial skill that can be applied to a wide range of problems.