Solve The Inequality:${ X^2 \leq 2x }$

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. They are used to describe the relationship between different quantities and are essential in solving various mathematical problems. In this article, we will focus on solving the inequality $x^2 \leq 2x$, which is a quadratic inequality. We will use algebraic methods to solve this inequality and provide a step-by-step solution.

Understanding the Inequality

The given inequality is $x^2 \leq 2x$. This is a quadratic inequality, where the left-hand side is a quadratic expression and the right-hand side is a linear expression. To solve this inequality, we need to find the values of $x$ that satisfy the given condition.

Step 1: Rearrange the Inequality

To solve the inequality, we can start by rearranging it to get all the terms on one side. We can subtract $2x$ from both sides of the inequality to get:

$x^2 - 2x \leq 0$

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the left-hand side. We can factor out $x$ from both terms to get:

$x(x - 2) \leq 0$

Step 3: Find the Critical Points

To solve the inequality, we need to find the critical points, which are the values of $x$ that make the expression $x(x - 2)$ equal to zero. We can set each factor equal to zero and solve for $x$ to get:

$x = 0$ and $x = 2$

Step 4: Test the Intervals

Now that we have the critical points, we need to test the intervals between them to see which ones satisfy the inequality. We can choose a test point from each interval and plug it into the inequality to see if it is true or false.

Interval 1: $(-\infty, 0)$

Let's choose a test point, say $x = -1$. Plugging this value into the inequality, we get:

$(-1)(-1 - 2) \leq 0$

Simplifying, we get:

$3 \leq 0$

This is false, so the interval $(-\infty, 0)$ does not satisfy the inequality.

Interval 2: $(0, 2)$

Let's choose a test point, say $x = 1$. Plugging this value into the inequality, we get:

$(1)(1 - 2) \leq 0$

Simplifying, we get:

$-1 \leq 0$

This is true, so the interval $(0, 2)$ satisfies the inequality.

Interval 3: $(2, \infty)$

Let's choose a test point, say $x = 3$. Plugging this value into the inequality, we get:

$(3)(3 - 2) \leq 0$

Simplifying, we get:

$3 \leq 0$

This is false, so the interval $(2, \infty)$ does not satisfy the inequality.

Step 5: Write the Solution

Based on the intervals that satisfy the inequality, we can write the solution as:

$0 \leq x \leq 2$

This is the solution to the inequality $x^2 \leq 2x$.

Conclusion

In this article, we solved the inequality $x^2 \leq 2x$ using algebraic methods. We rearranged the inequality, factored the quadratic expression, found the critical points, and tested the intervals to find the solution. The solution to the inequality is $0 \leq x \leq 2$. This is a fundamental concept in mathematics, and understanding how to solve inequalities is essential in solving various mathematical problems.

Frequently Asked Questions

• What is a quadratic inequality? A quadratic inequality is an inequality that involves a quadratic expression.
• How do you solve a quadratic inequality? To solve a quadratic inequality, you need to factor the quadratic expression, find the critical points, and test the intervals to find the solution.
• What are the critical points of a quadratic inequality? The critical points of a quadratic inequality are the values of $x$ that make the expression equal to zero.
• How do you test the intervals of a quadratic inequality? To test the intervals of a quadratic inequality, you need to choose a test point from each interval and plug it into the inequality to see if it is true or false.

Final Answer

The final answer is: $\boxed{0 \leq x \leq 2}$

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Introduction

In our previous article, we solved the inequality $x^2 \leq 2x$ using algebraic methods. We rearranged the inequality, factored the quadratic expression, found the critical points, and tested the intervals to find the solution. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving quadratic inequalities.

Q&A

Q: What is a quadratic inequality?

A: A quadratic inequality is an inequality that involves a quadratic expression. It is a mathematical statement that compares two expressions, one of which is a quadratic expression.

Q: How do you solve a quadratic inequality?

A: To solve a quadratic inequality, you need to follow these steps:

1. Rearrange the inequality to get all the terms on one side.
2. Factor the quadratic expression, if possible.
3. Find the critical points, which are the values of $x$ that make the expression equal to zero.
4. Test the intervals between the critical points to see which ones satisfy the inequality.

Q: What are the critical points of a quadratic inequality?

A: The critical points of a quadratic inequality are the values of $x$ that make the expression equal to zero. They are the points where the graph of the quadratic expression intersects the x-axis.

Q: How do you test the intervals of a quadratic inequality?

A: To test the intervals of a quadratic inequality, you need to choose a test point from each interval and plug it into the inequality to see if it is true or false. If the inequality is true for a test point, then the entire interval is part of the solution.

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

A: A quadratic equation is an equation that involves a quadratic expression, whereas a quadratic inequality is a statement that compares two expressions, one of which is a quadratic expression. In other words, a quadratic equation is an equation that is true for a specific value of $x$, whereas a quadratic inequality is a statement that is true for a range of values of $x$.

Q: Can you provide an example of a quadratic inequality?

A: Yes, here is an example of a quadratic inequality:

$x^2 + 4x + 4 \leq 0$

To solve this inequality, you would follow the steps outlined above.

Q: How do you graph a quadratic inequality?

A: To graph a quadratic inequality, you need to graph the quadratic expression and then shade the region that satisfies the inequality. The graph of the quadratic expression is a parabola, and the shaded region is the area where the inequality is true.

Q: Can you provide a real-world example of a quadratic inequality?

A: Yes, here is a real-world example of a quadratic inequality:

A company produces a certain product, and the cost of producing $x$ units of the product is given by the quadratic expression $x^2 + 4x + 4$. The company wants to know the range of values of $x$ for which the cost is less than or equal to $0$. This is a quadratic inequality, and solving it would give the company the desired information.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on solving quadratic inequalities. We hope that this article has been helpful in understanding the concept of quadratic inequalities and how to solve them.

Frequently Asked Questions

• What is a quadratic inequality?
• How do you solve a quadratic inequality?
• What are the critical points of a quadratic inequality?
• How do you test the intervals of a quadratic inequality?
• What is the difference between a quadratic inequality and a quadratic equation?
• Can you provide an example of a quadratic inequality?
• How do you graph a quadratic inequality?
• Can you provide a real-world example of a quadratic inequality?

Final Answer

The final answer is: $\boxed{0 \leq x \leq 2}$