Solve The Inequality: ${ |2x - 5| \leq 2 }$

Introduction

Inequalities with absolute values can be challenging to solve, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease. In this article, we will focus on solving the inequality $|2x - 5| \leq 2$. We will break down the solution into manageable steps, providing a clear explanation of each step and using examples to illustrate the concept.

Understanding Absolute Values

Before we dive into solving the inequality, let's take a moment to understand what absolute values are. The absolute value of a number is its distance from zero on the number line. In other words, it is the value of the number without considering its sign. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

The Inequality $|2x - 5| \leq 2$

The given inequality is $|2x - 5| \leq 2$. This means that the distance between $2x - 5$ and zero is less than or equal to 2. To solve this inequality, we need to find the values of $x$ that satisfy this condition.

Step 1: Setting Up the Inequality

To solve the inequality, we need to set up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative.

• Case 1: $2x - 5 \geq 0$
• Case 2: $2x - 5 < 0$

Step 2: Solving Case 1

For Case 1, we have $2x - 5 \geq 0$. To solve this inequality, we need to isolate the variable $x$.

from sympy import symbols, Eq, solve
x = symbols('x')

ineq = Eq(2*x - 5, 0)

solution = solve(ineq, x)
print(solution)

The solution to this inequality is $x \geq \frac{5}{2}$.

Step 3: Solving Case 2

For Case 2, we have $2x - 5 < 0$. To solve this inequality, we need to isolate the variable $x$.

from sympy import symbols, Eq, solve

x = symbols('x')

ineq = Eq(2*x - 5, 0)

solution = solve(ineq, x)
print(solution)

The solution to this inequality is $x < \frac{5}{2}$.

Step 4: Combining the Solutions

Now that we have solved both cases, we need to combine the solutions to find the final answer.

The solution to the inequality $|2x - 5| \leq 2$ is $\frac{5}{2} - 2 \leq x \leq \frac{5}{2} + 2$.

Conclusion

Solving inequalities with absolute values requires a clear understanding of the concept and a step-by-step approach. By breaking down the solution into manageable steps and using examples to illustrate the concept, we can tackle even the most challenging inequalities with ease. In this article, we solved the inequality $|2x - 5| \leq 2$ by setting up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative. We then solved each inequality and combined the solutions to find the final answer.

Final Answer

The final answer to the inequality $|2x - 5| \leq 2$ is $\frac{5}{2} - 2 \leq x \leq \frac{5}{2} + 2$.

Additional Resources

For more information on solving inequalities with absolute values, check out the following resources:

• Khan Academy: Solving Inequalities with Absolute Values
• Mathway: Solving Inequalities with Absolute Values
• Wolfram Alpha: Solving Inequalities with Absolute Values

FAQs

• Q: What is the absolute value of a number? A: The absolute value of a number is its distance from zero on the number line.
• Q: How do I solve an inequality with an absolute value? A: To solve an inequality with an absolute value, you need to set up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative.
• Q: What is the final answer to the inequality $|2x - 5| \leq 2$? A: The final answer to the inequality $|2x - 5| \leq 2$ is $\frac{5}{2} - 2 \leq x \leq \frac{5}{2} + 2$.
  Solving Inequalities with Absolute Values: A Q&A Guide ===========================================================

Introduction

In our previous article, we discussed how to solve inequalities with absolute values. However, we know that sometimes, the best way to learn is through asking questions and getting answers. In this article, we will provide a Q&A guide on solving inequalities with absolute values, covering common questions and topics.

Q: What is the absolute value of a number?

A: The absolute value of a number is its distance from zero on the number line. In other words, it is the value of the number without considering its sign. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

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

A: To solve an inequality with an absolute value, you need to set up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative. For example, if you have the inequality $|2x - 5| \leq 2$, you would set up two separate inequalities:

• Case 1: $2x - 5 \geq 0$
• Case 2: $2x - 5 < 0$

Q: What is the difference between $|x| \leq 2$ and $-2 \leq x \leq 2$?

A: The inequality $|x| \leq 2$ means that the distance between $x$ and zero is less than or equal to 2. This can be represented graphically as a closed interval on the number line, including the endpoints.

On the other hand, the inequality $-2 \leq x \leq 2$ means that $x$ is between -2 and 2, inclusive. This can also be represented graphically as a closed interval on the number line, including the endpoints.

Q: How do I solve the inequality $|x - 3| > 2$?

A: To solve the inequality $|x - 3| > 2$, you need to set up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative.

• Case 1: $x - 3 > 2$
• Case 2: $x - 3 < -2$

Solving these inequalities, you get:

• Case 1: $x > 5$
• Case 2: $x < 1$

Q: What is the final answer to the inequality $|2x - 5| \leq 2$?

A: The final answer to the inequality $|2x - 5| \leq 2$ is $\frac{5}{2} - 2 \leq x \leq \frac{5}{2} + 2$.

Q: How do I graph the inequality $|x| \leq 2$?

A: To graph the inequality $|x| \leq 2$, you need to draw a closed interval on the number line, including the endpoints. The interval should be centered at zero and extend 2 units in both directions.

Q: What is the difference between a closed interval and an open interval?

A: A closed interval is an interval that includes the endpoints, while an open interval is an interval that does not include the endpoints.

For example, the interval $[0, 2]$ is a closed interval, while the interval $(0, 2)$ is an open interval.

Conclusion

Solving inequalities with absolute values can be challenging, but with practice and patience, you can master the concept. In this article, we provided a Q&A guide on solving inequalities with absolute values, covering common questions and topics. We hope that this guide has been helpful in clarifying any doubts you may have had.

Additional Resources

For more information on solving inequalities with absolute values, check out the following resources:

• Khan Academy: Solving Inequalities with Absolute Values
• Mathway: Solving Inequalities with Absolute Values
• Wolfram Alpha: Solving Inequalities with Absolute Values

FAQs

• Q: What is the absolute value of a number? A: The absolute value of a number is its distance from zero on the number line.
• Q: How do I solve an inequality with an absolute value? A: To solve an inequality with an absolute value, you need to set up two separate inequalities, one for each case when the expression inside the absolute value is positive or negative.
• Q: What is the difference between $|x| \leq 2$ and $-2 \leq x \leq 2$? A: The inequality $|x| \leq 2$ means that the distance between $x$ and zero is less than or equal to 2, while the inequality $-2 \leq x \leq 2$ means that $x$ is between -2 and 2, inclusive.