Solve The Inequality.$8 \geq 3x - 13$

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Introduction to Inequalities

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Inequalities are mathematical expressions that compare two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. Solving inequalities is an essential skill in mathematics, and it has numerous applications in real-life situations.

What is a Linear Inequality?

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A linear inequality is an inequality that can be written in the form of a linear equation. It is an inequality that involves a linear expression, which is an expression that can be written in the form of ax + b, where a and b are constants, and x is the variable. For example, the inequality 2x + 3 ≥ 5 is a linear inequality because it can be written in the form of a linear equation.

Steps to Solve a Linear Inequality

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Solving a linear inequality involves several steps. Here are the steps to solve a linear inequality:

Step 1: Write the Inequality in the Form of a Linear Equation

The first step in solving a linear inequality is to write it in the form of a linear equation. This involves isolating the variable on one side of the inequality sign. For example, the inequality 2x + 3 ≥ 5 can be written as 2x ≥ 2.

Step 2: Add or Subtract the Same Value to Both Sides of the Inequality

The next step is to add or subtract the same value to both sides of the inequality. This will help to isolate the variable further. For example, if we have the inequality 2x ≥ 2, we can add 3 to both sides to get 2x + 3 ≥ 5.

Step 3: Multiply or Divide Both Sides of the Inequality by the Same Value

The next step is to multiply or divide both sides of the inequality by the same value. This will help to isolate the variable further. For example, if we have the inequality 2x + 3 ≥ 5, we can divide both sides by 2 to get x + 3/2 ≥ 5/2.

Step 4: Simplify the Inequality

The final step is to simplify the inequality. This involves combining like terms and eliminating any fractions. For example, if we have the inequality x + 3/2 ≥ 5/2, we can simplify it to x ≥ 1.

Solving the Inequality $8 \geq 3x - 13$

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Now that we have learned the steps to solve a linear inequality, let's apply them to the inequality $8 \geq 3x - 13$. To solve this inequality, we will follow the steps outlined above.

Step 1: Write the Inequality in the Form of a Linear Equation

The first step is to write the inequality in the form of a linear equation. This involves isolating the variable on one side of the inequality sign. We can do this by adding 13 to both sides of the inequality to get $8 + 13 \geq 3x - 13 + 13$, which simplifies to $21 \geq 3x$.

Step 2: Divide Both Sides of the Inequality by 3

The next step is to divide both sides of the inequality by 3. This will help to isolate the variable further. We can do this by dividing both sides of the inequality by 3 to get $21/3 \geq 3x/3$, which simplifies to $7 \geq x$.

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. This involves writing the solution as an interval, which is a set of values that the variable can take. In this case, the solution is $x \leq 7$, which can be written in interval notation as $(-\infty, 7]$.

Conclusion

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Solving inequalities is an essential skill in mathematics, and it has numerous applications in real-life situations. In this article, we have learned the steps to solve a linear inequality, and we have applied them to the inequality $8 \geq 3x - 13$. We have learned that the solution to this inequality is $x \leq 7$, which can be written in interval notation as $(-\infty, 7]$. We hope that this article has provided you with a better understanding of how to solve linear inequalities and has given you the confidence to tackle more complex inequalities in the future.

Frequently Asked Questions

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation. It is an inequality that involves a linear expression, which is an expression that can be written in the form of ax + b, where a and b are constants, and x is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow the steps outlined above. First, write the inequality in the form of a linear equation. Then, add or subtract the same value to both sides of the inequality. Next, multiply or divide both sides of the inequality by the same value. Finally, simplify the inequality.

Q: What is the solution to the inequality $8 \geq 3x - 13$?

A: The solution to the inequality $8 \geq 3x - 13$ is $x \leq 7$, which can be written in interval notation as $(-\infty, 7]$.

References

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• [1] "Algebra and Trigonometry" by James Stewart
• [2] "College Algebra" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

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• Khan Academy: Linear Inequalities
• Mathway: Linear Inequalities
• Wolfram Alpha: Linear Inequalities
  

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Introduction

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Solving linear inequalities can be a challenging task, but with the right guidance, it can be made easier. In this article, we will answer some of the most frequently asked questions about solving linear inequalities.

Q&A

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation. It is an inequality that involves a linear expression, which is an expression that can be written in the form of ax + b, where a and b are constants, and x is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow the steps outlined below:

1. Write the inequality in the form of a linear equation.
2. Add or subtract the same value to both sides of the inequality.
3. Multiply or divide both sides of the inequality by the same value.
4. Simplify 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 of a linear equation, while a quadratic inequality is an inequality that can be written in the form of a quadratic equation. Quadratic inequalities are more complex and require different techniques to solve.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to follow the steps outlined below:

1. Write the inequality in the form of a quadratic equation.
2. Factor the quadratic expression.
3. Set each factor equal to zero and solve for x.
4. Use a number line or a graph to determine the solution set.

Q: What is the solution to the inequality $8 \geq 3x - 13$?

A: The solution to the inequality $8 \geq 3x - 13$ is $x \leq 7$, which can be written in interval notation as $(-\infty, 7]$.

Q: How do I write the solution to an inequality in interval notation?

A: To write the solution to an inequality in interval notation, you need to follow the steps outlined below:

1. Determine the direction of the inequality.
2. Write the solution set in the form of an interval.
3. Use the correct notation to indicate the direction of the inequality.

Q: What is the difference between a discrete and a continuous interval?

A: A discrete interval is an interval that consists of a finite number of points, while a continuous interval is an interval that consists of an infinite number of points.

Q: How do I determine the direction of an inequality?

A: To determine the direction of an inequality, you need to follow the steps outlined below:

1. Look at the inequality sign.
2. Determine the direction of the inequality.
3. Use the correct notation to indicate the direction of the inequality.

Conclusion

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Solving linear inequalities can be a challenging task, but with the right guidance, it can be made easier. In this article, we have answered some of the most frequently asked questions about solving linear inequalities. We hope that this article has provided you with a better understanding of how to solve linear inequalities and has given you the confidence to tackle more complex inequalities in the future.

Frequently Asked Questions (FAQs)

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Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of a linear equation.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to follow the steps outlined above.

Q: What is the solution to the inequality $8 \geq 3x - 13$?

A: The solution to the inequality $8 \geq 3x - 13$ is $x \leq 7$, which can be written in interval notation as $(-\infty, 7]$.

Q: How do I write the solution to an inequality in interval notation?

A: To write the solution to an inequality in interval notation, you need to follow the steps outlined above.

References

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• [1] "Algebra and Trigonometry" by James Stewart
• [2] "College Algebra" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

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• Khan Academy: Linear Inequalities
• Mathway: Linear Inequalities
• Wolfram Alpha: Linear Inequalities