Choose The Missing Step In The Given Solution To The Inequality $-2x - 14 \ \textgreater \ -4 + 3x$.$\[ \begin{array}{l} -2x - 14 \ \textgreater \ -4 + 3x \\ -5x - 14 \ \textgreater \ -4 \\ x \ \textless \

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Introduction

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Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving linear inequalities, specifically the inequality $-2x - 14 \ \textgreater \ -4 + 3x$. We will break down the solution step-by-step and identify the missing step in the given solution.

Understanding the Inequality

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The given inequality is $-2x - 14 \ \textgreater \ -4 + 3x$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to add $2x$ to both sides of the inequality to get rid of the negative term.

Step 1: Add 2x to both sides

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$$-2x - 14 \ \textgreater \ -4 + 3x$$

$$-2x + 2x - 14 \ \textgreater \ -4 + 3x + 2x$$

$$-14 \ \textgreater \ -4 + 5x$$

Simplifying the Inequality

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The next step is to simplify the inequality by combining like terms. We can start by adding $14$ to both sides of the inequality to get rid of the negative term.

Step 2: Add 14 to both sides

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$$-14 \ \textgreater \ -4 + 5x$$

$$-14 + 14 \ \textgreater \ -4 + 5x + 14$$

$$0 \ \textgreater \ 10 + 5x$$

Isolating the Variable

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The next step is to isolate the variable $x$ on one side of the inequality. We can start by subtracting $10$ from both sides of the inequality to get rid of the constant term.

Step 3: Subtract 10 from both sides

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$$0 \ \textgreater \ 10 + 5x$$

$$0 - 10 \ \textgreater \ 10 + 5x - 10$$

$$-10 \ \textgreater \ 5x$$

Dividing by 5

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The final step is to divide both sides of the inequality by $5$ to solve for $x$.

Step 4: Divide by 5

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$$-10 \ \textgreater \ 5x$$

$$\frac{-10}{5} \ \textgreater \ \frac{5x}{5}$$

$$-2 \ \textgreater \ x$$

Conclusion

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In conclusion, the missing step in the given solution to the inequality $-2x - 14 \ \textgreater \ -4 + 3x$ is to divide both sides of the inequality by $5$ to solve for $x$. The correct solution is $x \ \textless \ -2$.

Discussion

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Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. In this article, we have broken down the solution to the inequality $-2x - 14 \ \textgreater \ -4 + 3x$ step-by-step, highlighting the importance of isolating the variable on one side of the inequality sign. By following these steps, students can develop a strong foundation in solving linear inequalities and apply this knowledge to a wide range of mathematical problems.

Common Mistakes

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When solving inequalities, students often make common mistakes, such as:

• Not isolating the variable: Failing to isolate the variable on one side of the inequality sign can lead to incorrect solutions.
• Not maintaining the direction of the inequality: Changing the direction of the inequality sign can result in incorrect solutions.
• Not considering the sign of the coefficient: Failing to consider the sign of the coefficient of the variable can lead to incorrect solutions.

Tips and Tricks

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To avoid common mistakes and ensure accurate solutions, students can follow these tips and tricks:

• Read the inequality carefully: Read the inequality carefully to ensure that you understand the direction of the inequality sign and the sign of the coefficient.
• Isolate the variable: Isolate the variable on one side of the inequality sign to ensure accurate solutions.
• Maintain the direction of the inequality: Maintain the direction of the inequality sign to ensure accurate solutions.
• Consider the sign of the coefficient: Consider the sign of the coefficient of the variable to ensure accurate solutions.

Real-World Applications

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Solving inequalities has numerous real-world applications, including:

• Finance: In finance, inequalities are used to model financial transactions and investments.
• Science: In science, inequalities are used to model physical systems and phenomena.
• Engineering: In engineering, inequalities are used to design and optimize systems.

Conclusion

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In conclusion, solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. By following the steps outlined in this article, students can develop a strong foundation in solving linear inequalities and apply this knowledge to a wide range of mathematical problems. Remember to read the inequality carefully, isolate the variable, maintain the direction of the inequality, and consider the sign of the coefficient to ensure accurate solutions.

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Introduction

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In our previous article, we discussed the step-by-step solution to the inequality $-2x - 14 \ \textgreater \ -4 + 3x$. We also highlighted the importance of isolating the variable on one side of the inequality sign and maintaining the direction of the inequality. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

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

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A: A linear inequality is an inequality that can be written in the form $ax + b \ \textgreater \ c$, 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 $ax^2 + bx + c \ \textgreater \ d$, where $a$, $b$, $c$, and $d$ are constants and $x$ is the variable.

Q: How do I solve a quadratic inequality?

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A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression: Factor the quadratic expression on the left-hand side of the inequality.
2. Set each factor equal to zero: Set each factor equal to zero and solve for $x$.
3. Graph the solutions: Graph the solutions on a number line.
4. Determine the solution set: Determine the solution set by shading the appropriate regions on the number line.

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

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A: A strict inequality is an inequality that uses the symbols $>$ or $<$, while a non-strict inequality is an inequality that uses the symbols $\geq$ or $\leq$.

Q: How do I solve a system of linear inequalities?

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A: To solve a system of linear inequalities, you can use the following steps:

1. Graph each inequality: Graph each inequality on a coordinate plane.
2. Find the intersection: Find the intersection of the two inequalities.
3. Determine the solution set: Determine the solution set by shading the appropriate regions on the coordinate plane.

Q: What is the importance of solving inequalities in real-world applications?

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A: Solving inequalities is essential in real-world applications, such as finance, science, and engineering. In finance, inequalities are used to model financial transactions and investments. In science, inequalities are used to model physical systems and phenomena. In engineering, inequalities are used to design and optimize systems.

Conclusion

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In conclusion, solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. By following the steps outlined in this article, students can develop a strong foundation in solving linear inequalities and apply this knowledge to a wide range of mathematical problems. Remember to read the inequality carefully, isolate the variable, maintain the direction of the inequality, and consider the sign of the coefficient to ensure accurate solutions.

Common Mistakes

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When solving inequalities, students often make common mistakes, such as:

• Not isolating the variable: Failing to isolate the variable on one side of the inequality sign can lead to incorrect solutions.
• Not maintaining the direction of the inequality: Changing the direction of the inequality sign can result in incorrect solutions.
• Not considering the sign of the coefficient: Failing to consider the sign of the coefficient of the variable can lead to incorrect solutions.

Tips and Tricks

---

To avoid common mistakes and ensure accurate solutions, students can follow these tips and tricks:

• Read the inequality carefully: Read the inequality carefully to ensure that you understand the direction of the inequality sign and the sign of the coefficient.
• Isolate the variable: Isolate the variable on one side of the inequality sign to ensure accurate solutions.
• Maintain the direction of the inequality: Maintain the direction of the inequality sign to ensure accurate solutions.
• Consider the sign of the coefficient: Consider the sign of the coefficient of the variable to ensure accurate solutions.

Real-World Applications

---

Solving inequalities has numerous real-world applications, including:

• Finance: In finance, inequalities are used to model financial transactions and investments.
• Science: In science, inequalities are used to model physical systems and phenomena.
• Engineering: In engineering, inequalities are used to design and optimize systems.

Conclusion

---

In conclusion, solving inequalities is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. By following the steps outlined in this article, students can develop a strong foundation in solving linear inequalities and apply this knowledge to a wide range of mathematical problems. Remember to read the inequality carefully, isolate the variable, maintain the direction of the inequality, and consider the sign of the coefficient to ensure accurate solutions.