Solve The Inequality: $\[ -\frac{8}{3} Z + 2z - 6 \leq 3z + 5 \\]

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Introduction

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In this article, we will delve into the world of inequalities and learn how to solve a specific one. The given inequality is ${ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 }$. We will break down the solution process into manageable steps, making it easy to understand and follow along.

Understanding the Inequality

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Before we dive into the solution, let's take a closer look at the inequality. The given inequality is a linear inequality, which means it can be written in the form of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants.

In this case, the inequality is ${ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 }$. Our goal is to isolate the variable $z$ and find the solution set that satisfies the inequality.

Step 1: Combine Like Terms

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The first step in solving the inequality is to combine like terms. We can start by combining the terms with the variable $z$.

{ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 \}

Combine the terms with $z$:

{ -\frac{8}{3} z + 2z = -\frac{8}{3} z + \frac{6}{3} z = -\frac{2}{3} z \}

Now, the inequality becomes:

{ -\frac{2}{3} z - 6 \leq 3z + 5 \}

Step 2: Add 6 to Both Sides

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The next step is to add 6 to both sides of the inequality to get rid of the negative term.

{ -\frac{2}{3} z - 6 \leq 3z + 5 \}

Add 6 to both sides:

{ -\frac{2}{3} z \leq 3z + 11 \}

Step 3: Subtract 3z from Both Sides

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Now, we need to subtract 3z from both sides of the inequality to isolate the term with $z$.

{ -\frac{2}{3} z \leq 3z + 11 \}

Subtract 3z from both sides:

{ -\frac{2}{3} z - 3z \leq 11 \}

Combine the terms with $z$:

{ -\frac{11}{3} z \leq 11 \}

Step 4: Multiply Both Sides by -3

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To get rid of the fraction, we need to multiply both sides of the inequality by -3.

{ -\frac{11}{3} z \leq 11 \}

Multiply both sides by -3:

{ 11z \geq -33 \}

Step 5: Divide Both Sides by 11

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Finally, we need to divide both sides of the inequality by 11 to solve for $z$.

{ 11z \geq -33 \}

Divide both sides by 11:

{ z \geq -3 \}

Conclusion

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In this article, we solved the inequality ${ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 }$. We broke down the solution process into manageable steps, making it easy to understand and follow along.

The final solution is ${ z \geq -3 }$. This means that the solution set is all real numbers greater than or equal to -3.

Frequently Asked Questions

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Q: What is the solution set for the inequality?

A: The solution set is all real numbers greater than or equal to -3.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by a non-zero 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 of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants. A quadratic inequality is an inequality that can be written in the form of $ax^2 + bx + c \leq dx^2 + ex + f$, where $a$, $b$, $c$, $d$, $e$, and $f$ are constants.

Final Thoughts

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Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to break down the solution process into manageable steps, and don't be afraid to ask for help if you get stuck.

In this article, we solved the inequality ${ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 }$. We hope that this article has provided you with a clear understanding of how to solve linear inequalities and has given you the confidence to tackle more complex inequalities in the future.

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Introduction

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In our previous article, we solved the inequality ${ -\frac{8}{3} z + 2z - 6 \leq 3z + 5 }$. We broke down the solution process into manageable steps, making it easy to understand and follow along.

In this article, we will provide a Q&A guide to help you better understand how to solve inequalities. We will cover common questions and topics related to solving inequalities, including linear inequalities, quadratic inequalities, and more.

Q&A: Solving Inequalities

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

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by a non-zero value.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to combine like terms. This involves grouping together terms with the same variable and coefficient.

Q: How do I know which direction to move the inequality sign when adding or subtracting the same value from both sides?

A: When adding or subtracting the same value from both sides of an inequality, you need to move the inequality sign in the opposite direction. For example, if you have the inequality $x + 3 \leq 5$, you would subtract 3 from both sides to get $x \leq 2$.

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

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

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 quadratic expression to determine the solution set.

Q: What is the significance of the discriminant in solving quadratic inequalities?

A: The discriminant is the value under the square root in the quadratic formula. It can be used to determine the nature of the solutions to a quadratic equation.

Q&A: Graphing Inequalities

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Q: What is the difference between a linear inequality and a quadratic inequality in terms of their graphs?

A: A linear inequality can be represented by a straight line on a graph, while a quadratic inequality can be represented by a parabola.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality.

Q: How do I graph a quadratic inequality?

A: To graph a quadratic inequality, you need to graph the corresponding quadratic equation and then shade the region that satisfies the inequality.

Q&A: Advanced Topics

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

A: A linear inequality is an inequality that can be written in the form of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants. A nonlinear inequality is an inequality that cannot be written in this form.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to use advanced techniques such as substitution, elimination, or graphing.

Q: What is the significance of the concept of "feasible region" in solving inequalities?

A: The feasible region is the set of all possible solutions to an inequality. It can be used to determine the solution set of an inequality.

Conclusion

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In this article, we provided a Q&A guide to help you better understand how to solve inequalities. We covered common questions and topics related to solving inequalities, including linear inequalities, quadratic inequalities, and more.

We hope that this article has provided you with a clear understanding of how to solve inequalities and has given you the confidence to tackle more complex inequalities in the future.

Final Thoughts

---

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to break down the solution process into manageable steps, and don't be afraid to ask for help if you get stuck.

In this article, we covered a range of topics related to solving inequalities, from linear inequalities to quadratic inequalities and more. We hope that this article has provided you with a comprehensive understanding of how to solve inequalities and has given you the confidence to tackle more complex inequalities in the future.

Frequently Asked Questions

---

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

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by a non-zero value.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to combine like terms. This involves grouping together terms with the same variable and coefficient.

Q: How do I know which direction to move the inequality sign when adding or subtracting the same value from both sides?

A: When adding or subtracting the same value from both sides of an inequality, you need to move the inequality sign in the opposite direction. For example, if you have the inequality $x + 3 \leq 5$, you would subtract 3 from both sides to get $x \leq 2$.

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

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

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 quadratic expression to determine the solution set.

Q: What is the significance of the discriminant in solving quadratic inequalities?

A: The discriminant is the value under the square root in the quadratic formula. It can be used to determine the nature of the solutions to a quadratic equation.

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

A: A linear inequality is an inequality that can be written in the form of $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants. A nonlinear inequality is an inequality that cannot be written in this form.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you need to use advanced techniques such as substitution, elimination, or graphing.

Q: What is the significance of the concept of "feasible region" in solving inequalities?

A: The feasible region is the set of all possible solutions to an inequality. It can be used to determine the solution set of an inequality.