Solve The Inequality:$\[ 5x - 15 \leq 5 \\]

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Introduction

Inequalities are mathematical expressions that compare two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $5x - 15 \leq 5$.

Understanding the Inequality

The given inequality is $5x - 15 \leq 5$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to add 15 to both sides of the inequality to get rid of the negative term.

Adding 15 to Both Sides

When we add 15 to both sides of the inequality, we get:

${ 5x - 15 + 15 \leq 5 + 15 }$

This simplifies to:

${ 5x \leq 20 }$

Isolating the Variable

Now that we have $5x \leq 20$, we need to isolate the variable $x$ on one side of the inequality sign. To do this, we divide both sides of the inequality by 5.

Dividing Both Sides by 5

When we divide both sides of the inequality by 5, we get:

${ \frac{5x}{5} \leq \frac{20}{5} }$

This simplifies to:

${ x \leq 4 }$

Conclusion

In conclusion, the solution to the inequality $5x - 15 \leq 5$ is $x \leq 4$. This means that any value of $x$ that is less than or equal to 4 is a solution to the inequality.

Graphical Representation

To visualize the solution to the inequality, we can graph the inequality on a number line. The number line represents all possible values of $x$. The inequality $x \leq 4$ means that all values of $x$ to the left of 4 are included in the solution.

Graphing the Inequality

To graph the inequality, we draw a closed circle at $x = 4$ to indicate that this value is included in the solution. We then draw an arrow to the left of $x = 4$ to indicate that all values to the left of 4 are included in the solution.

Real-World Applications

Inequalities have many real-world applications. For example, in finance, inequalities are used to calculate interest rates and investment returns. In engineering, inequalities are used to design and optimize systems. In medicine, inequalities are used to model and analyze the spread of diseases.

Example 1: Calculating Interest Rates

Suppose we want to calculate the interest rate on a savings account. The interest rate is given by the inequality $r \leq 5\%$. This means that the interest rate is less than or equal to 5%.

Example 2: Designing a System

Suppose we want to design a system that can handle a maximum of 1000 units of data. The system is given by the inequality $x \leq 1000$. This means that the system can handle any value of $x$ that is less than or equal to 1000.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always add or subtract the same value to both sides of the inequality.
• Always multiply or divide both sides of the inequality by the same value.
• Always check the direction of the inequality sign.
• Always graph the inequality on a number line to visualize the solution.

Conclusion

In conclusion, solving inequalities involves isolating the variable on one side of the inequality sign. The given inequality $5x - 15 \leq 5$ is solved by adding 15 to both sides, then dividing both sides by 5. The solution to the inequality is $x \leq 4$. Inequalities have many real-world applications, and it's essential to remember the tips and tricks for solving inequalities.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This involves adding or subtracting the same value to both sides of the inequality, then multiplying or dividing both sides by the same 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 $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to draw a closed circle at the value that is included in the solution, then draw an arrow to the left or right of the value to indicate the direction of the inequality.

References

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Mathway
• [3] "Graphing Inequalities" by Purplemath
  

Introduction

In our previous article, we discussed how to solve the inequality $5x - 15 \leq 5$. In this article, we will provide a Q&A guide to help you understand and solve inequalities.

Q&A Guide

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using a relation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This involves adding or subtracting the same value to both sides of the inequality, then multiplying or dividing both sides by the same 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 $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to draw a closed circle at the value that is included in the solution, then draw an arrow to the left or right of the value to indicate the direction of the inequality.

Q: What is the solution to the inequality $x + 2 \leq 5$?

A: To solve the inequality $x + 2 \leq 5$, we need to isolate the variable $x$ on one side of the inequality sign. We can do this by subtracting 2 from both sides of the inequality:

${ x + 2 - 2 \leq 5 - 2 }$

This simplifies to:

${ x \leq 3 }$

Q: What is the solution to the inequality $2x - 3 \geq 7$?

A: To solve the inequality $2x - 3 \geq 7$, we need to isolate the variable $x$ on one side of the inequality sign. We can do this by adding 3 to both sides of the inequality, then dividing both sides by 2:

${ 2x - 3 + 3 \geq 7 + 3 }$

This simplifies to:

${ 2x \geq 10 }$

Dividing both sides by 2, we get:

${ x \geq 5 }$

Q: How do I solve a compound inequality?

A: A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". To solve a compound inequality, you need to solve each inequality separately, then combine the solutions.

For example, consider the compound inequality $x > 2$ and $x < 5$. To solve this inequality, we need to solve each inequality separately:

${ x > 2 }$

This means that $x$ is greater than 2.

${ x < 5 }$

This means that $x$ is less than 5.

Combining the two inequalities, we get:

${ 2 < x < 5 }$

This means that $x$ is between 2 and 5, but not equal to 2 or 5.

Q: How do I solve an inequality with absolute value?

A: An inequality with absolute value is an inequality that involves the absolute value of a variable. To solve an inequality with absolute value, you need to consider two cases: one where the variable is positive, and one where the variable is negative.

For example, consider the inequality $|x| \leq 3$. To solve this inequality, we need to consider two cases:

Case 1: $x \geq 0$

In this case, the absolute value of $x$ is equal to $x$ itself. So, we can write the inequality as:

${ x \leq 3 }$

Case 2: $x < 0$

In this case, the absolute value of $x$ is equal to $-x$. So, we can write the inequality as:

${ -x \leq 3 }$

Multiplying both sides by -1, we get:

${ x \geq -3 }$

Combining the two cases, we get:

${ -3 \leq x \leq 3 }$

This means that $x$ is between -3 and 3, inclusive.

Conclusion

In conclusion, solving inequalities involves isolating the variable on one side of the inequality sign. We have discussed how to solve linear inequalities, quadratic inequalities, and compound inequalities. We have also discussed how to solve inequalities with absolute value. By following these steps, you can solve any inequality that comes your way.

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

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to draw a closed circle at the value that is included in the solution, then draw an arrow to the left or right of the value to indicate the direction of the inequality.

Q: What is the solution to the inequality $x + 2 \leq 5$?

A: To solve the inequality $x + 2 \leq 5$, we need to isolate the variable $x$ on one side of the inequality sign. We can do this by subtracting 2 from both sides of the inequality:

${ x + 2 - 2 \leq 5 - 2 }$

This simplifies to:

${ x \leq 3 }$

Q: How do I solve a compound inequality?

A: A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". To solve a compound inequality, you need to solve each inequality separately, then combine the solutions.

References

• [1] "Inequalities" by Khan Academy
• [2] "Solving Inequalities" by Mathway
• [3] "Graphing Inequalities" by Purplemath