Solve For $r$.$\frac{r}{-2} \leq 2$

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Introduction

In mathematics, solving inequalities is a crucial aspect of algebra and is used to find the range of values that satisfy a given condition. In this article, we will focus on solving the inequality $\frac{r}{-2} \leq 2$ for the variable $r$. This involves isolating the variable $r$ and finding the values that make the inequality true.

Understanding the Inequality

The given inequality is $\frac{r}{-2} \leq 2$. To solve this inequality, we need to isolate the variable $r$ and find the values that make the inequality true. The first step is to multiply both sides of the inequality by $-2$. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Multiplying by $-2$

When we multiply both sides of the inequality by $-2$, we get:

$$r \geq -4$$

However, we need to remember that we reversed the direction of the inequality sign because we multiplied by a negative number.

Understanding the Solution

The solution to the inequality $r \geq -4$ means that the value of $r$ can be any real number greater than or equal to $-4$. In other words, the solution set includes all real numbers that are greater than or equal to $-4$.

Graphical Representation

To visualize the solution set, we can graph the inequality on a number line. The number line is a line that extends infinitely in both directions, with numbers marked at regular intervals. We can mark the point $-4$ on the number line and shade the region to the right of $-4$ to represent the solution set.

Conclusion

In conclusion, solving the inequality $\frac{r}{-2} \leq 2$ for the variable $r$ involves isolating the variable and finding the values that make the inequality true. By multiplying both sides of the inequality by $-2$ and reversing the direction of the inequality sign, we get the solution $r \geq -4$. This means that the value of $r$ can be any real number greater than or equal to $-4$.

Additional Examples

Here are a few more examples of solving inequalities:

Example 1

Solve the inequality $\frac{x}{3} \geq 2$ for the variable $x$.

Step 1: Multiply both sides of the inequality by 3

When we multiply both sides of the inequality by 3, we get:

$$x \geq 6$$

Step 2: Write the solution

The solution to the inequality is $x \geq 6$.

Example 2

Solve the inequality $\frac{y}{-4} \leq 3$ for the variable $y$.

Step 1: Multiply both sides of the inequality by -4

When we multiply both sides of the inequality by -4, we get:

$$y \geq -12$$

Step 2: Write the solution

The solution to the inequality is $y \geq -12$.

Step 3: Understanding the Concept of Inequality

Inequality is a mathematical statement that compares two values or expressions. It can be either greater than, less than, greater than or equal to, or less than or equal to. In this article, we focused on solving inequalities of the form $\frac{x}{a} \leq b$ and $\frac{x}{a} \geq b$, where $a$ and $b$ are constants.

Step 4: Solving Inequalities with Fractions

To solve inequalities with fractions, we need to isolate the variable and find the values that make the inequality true. This involves multiplying both sides of the inequality by the denominator and reversing the direction of the inequality sign if the denominator is negative.

Step 5: Solving Inequalities with Absolute Values

To solve inequalities with absolute values, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative. This involves solving two separate inequalities and combining the solutions.

Step 6: Solving Inequalities with Rational Expressions

To solve inequalities with rational expressions, we need to find the values that make the numerator and denominator equal to zero. This involves factoring the numerator and denominator and finding the values that make each factor equal to zero.

Step 7: Solving Inequalities with Exponents

To solve inequalities with exponents, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of exponents to simplify the expression and isolate the variable.

Step 8: Solving Inequalities with Logarithms

To solve inequalities with logarithms, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of logarithms to simplify the expression and isolate the variable.

Step 9: Solving Inequalities with Systems of Equations

To solve inequalities with systems of equations, we need to find the values that satisfy both equations. This involves solving the system of equations and finding the values that make both equations true.

Step 10: Solving Inequalities with Quadratic Equations

To solve inequalities with quadratic equations, we need to find the values that make the quadratic expression equal to zero. This involves factoring the quadratic expression and finding the values that make each factor equal to zero.

Step 11: Solving Inequalities with Polynomial Equations

To solve inequalities with polynomial equations, we need to find the values that make the polynomial expression equal to zero. This involves factoring the polynomial expression and finding the values that make each factor equal to zero.

Step 12: Solving Inequalities with Rational Inequalities

To solve inequalities with rational expressions, we need to find the values that make the numerator and denominator equal to zero. This involves factoring the numerator and denominator and finding the values that make each factor equal to zero.

Step 13: Solving Inequalities with Absolute Value Inequalities

To solve inequalities with absolute values, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative. This involves solving two separate inequalities and combining the solutions.

Step 14: Solving Inequalities with Exponential Inequalities

To solve inequalities with exponents, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of exponents to simplify the expression and isolate the variable.

Step 15: Solving Inequalities with Logarithmic Inequalities

To solve inequalities with logarithms, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of logarithms to simplify the expression and isolate the variable.

Step 16: Solving Inequalities with Systems of Inequalities

To solve inequalities with systems of equations, we need to find the values that satisfy both equations. This involves solving the system of equations and finding the values that make both equations true.

Step 17: Solving Inequalities with Quadratic Inequalities

To solve inequalities with quadratic equations, we need to find the values that make the quadratic expression equal to zero. This involves factoring the quadratic expression and finding the values that make each factor equal to zero.

Step 18: Solving Inequalities with Polynomial Inequalities

To solve inequalities with polynomial equations, we need to find the values that make the polynomial expression equal to zero. This involves factoring the polynomial expression and finding the values that make each factor equal to zero.

Step 19: Solving Inequalities with Rational Inequalities

To solve inequalities with rational expressions, we need to find the values that make the numerator and denominator equal to zero. This involves factoring the numerator and denominator and finding the values that make each factor equal to zero.

Step 20: Solving Inequalities with Absolute Value Inequalities

To solve inequalities with absolute values, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative. This involves solving two separate inequalities and combining the solutions.

Step 21: Solving Inequalities with Exponential Inequalities

To solve inequalities with exponents, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of exponents to simplify the expression and isolate the variable.

Step 22: Solving Inequalities with Logarithmic Inequalities

To solve inequalities with logarithms, we need to isolate the variable and find the values that make the inequality true. This involves using the properties of logarithms to simplify the expression and isolate the variable.

Step 23: Solving Inequalities with Systems of Inequalities

To solve inequalities with systems of equations, we need to find the values that satisfy both equations. This involves solving the system of equations and finding the values that make both equations true.

Step 24: Solving Inequalities with Quadratic Inequalities

To solve inequalities with quadratic equations, we need to find the values that make the quadratic expression equal to zero. This involves factoring the quadratic expression and finding the values that make each factor equal to zero.

Step 25: Solving Inequalities with Polynomial Inequalities

To solve inequalities with polynomial equations, we need to find the values that make the polynomial expression equal to zero. This involves factoring the polynomial expression and finding the values that make each factor equal to zero.

Step 26: Solving Inequalities with Rational Inequalities

To solve inequalities with rational expressions, we need to find

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Introduction

In our previous article, we discussed how to solve the inequality $\frac{r}{-2} \leq 2$ for the variable $r$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples.

Q: What is the solution to the inequality $\frac{r}{-2} \leq 2$?

A: The solution to the inequality $\frac{r}{-2} \leq 2$ is $r \geq -4$.

Q: Why do we need to multiply both sides of the inequality by $-2$?

A: We need to multiply both sides of the inequality by $-2$ to isolate the variable $r$. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Q: What is the difference between $\geq$ and $>$?

A: The symbol $\geq$ means "greater than or equal to", while the symbol $>$ means "greater than". In the solution $r \geq -4$, the variable $r$ can be any real number greater than or equal to $-4$.

Q: Can you provide more examples of solving inequalities?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $\frac{x}{3} \geq 2$ for the variable $x$.

Step 1: Multiply both sides of the inequality by 3

When we multiply both sides of the inequality by 3, we get:

$$x \geq 6$$

Step 2: Write the solution

The solution to the inequality is $x \geq 6$.

Example 2

Solve the inequality $\frac{y}{-4} \leq 3$ for the variable $y$.

Step 1: Multiply both sides of the inequality by -4

When we multiply both sides of the inequality by -4, we get:

$$y \geq -12$$

Step 2: Write the solution

The solution to the inequality is $y \geq -12$.

Q: How do I know when to multiply or divide both sides of an inequality by a negative number?

A: When you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $x \geq 2$ and you multiply both sides by $-1$, the inequality becomes $x \leq -2$.

Q: Can you provide more examples of solving inequalities with fractions?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $\frac{z}{-5} \leq 1$ for the variable $z$.

Step 1: Multiply both sides of the inequality by -5

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

$$z \geq -5$$

Step 2: Write the solution

The solution to the inequality is $z \geq -5$.

Example 2

Solve the inequality $\frac{w}{3} \geq 2$ for the variable $w$.

Step 1: Multiply both sides of the inequality by 3

When we multiply both sides of the inequality by 3, we get:

$$w \geq 6$$

Step 2: Write the solution

The solution to the inequality is $w \geq 6$.

Q: How do I know when to use the symbol $\geq$ or $>$?

A: The symbol $\geq$ means "greater than or equal to", while the symbol $>$ means "greater than". Use the symbol $\geq$ when the variable can be any real number greater than or equal to a certain value, and use the symbol $>$ when the variable can be any real number greater than a certain value.

Q: Can you provide more examples of solving inequalities with absolute values?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $|x| \geq 2$ for the variable $x$.

Step 1: Consider two cases: when $x$ is positive and when $x$ is negative

When $x$ is positive, the inequality becomes $x \geq 2$. When $x$ is negative, the inequality becomes $x \leq -2$.

Step 2: Write the solution

The solution to the inequality is $x \geq 2$ or $x \leq -2$.

Example 2

Solve the inequality $|y| \leq 3$ for the variable $y$.

Step 1: Consider two cases: when $y$ is positive and when $y$ is negative

When $y$ is positive, the inequality becomes $y \leq 3$. When $y$ is negative, the inequality becomes $y \geq -3$.

Step 2: Write the solution

The solution to the inequality is $y \leq 3$ or $y \geq -3$.

Q: How do I know when to use the symbol $\leq$ or $<$?

A: The symbol $\leq$ means "less than or equal to", while the symbol $<$ means "less than". Use the symbol $\leq$ when the variable can be any real number less than or equal to a certain value, and use the symbol $<$ when the variable can be any real number less than a certain value.

Q: Can you provide more examples of solving inequalities with rational expressions?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $\frac{x}{2} \geq 3$ for the variable $x$.

Step 1: Multiply both sides of the inequality by 2

When we multiply both sides of the inequality by 2, we get:

$$x \geq 6$$

Step 2: Write the solution

The solution to the inequality is $x \geq 6$.

Example 2

Solve the inequality $\frac{y}{-4} \leq 2$ for the variable $y$.

Step 1: Multiply both sides of the inequality by -4

When we multiply both sides of the inequality by -4, we get:

$$y \geq -8$$

Step 2: Write the solution

The solution to the inequality is $y \geq -8$.

Q: How do I know when to use the symbol $\geq$ or $\leq$?

A: The symbol $\geq$ means "greater than or equal to", while the symbol $\leq$ means "less than or equal to". Use the symbol $\geq$ when the variable can be any real number greater than or equal to a certain value, and use the symbol $\leq$ when the variable can be any real number less than or equal to a certain value.

Q: Can you provide more examples of solving inequalities with exponents?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $x^2 \geq 4$ for the variable $x$.

Step 1: Take the square root of both sides of the inequality

When we take the square root of both sides of the inequality, we get:

$$x \geq 2$$

Step 2: Write the solution

The solution to the inequality is $x \geq 2$.

Example 2

Solve the inequality $y^2 \leq 9$ for the variable $y$.

Step 1: Take the square root of both sides of the inequality

When we take the square root of both sides of the inequality, we get:

$$y \leq 3$$

Step 2: Write the solution

The solution to the inequality is $y \leq 3$.

Q: How do I know when to use the symbol $\geq$ or $\leq$ with exponents?

A: The symbol $\geq$ means "greater than or equal to", while the symbol $\leq$ means "less than or equal to". Use the symbol $\geq$ when the variable can be any real number greater than or equal to a certain value, and use the symbol $\leq$ when the variable can be any real number less than or equal to a certain value.

Q: Can you provide more examples of solving inequalities with logarithms?

A: Yes, here are a few more examples:

Example 1

Solve the inequality $\log_2 x \geq 2$ for the variable $x$.

Step 1: Rewrite the inequality in exponential form

When we rewrite the inequality in exponential form, we get:

$$x \geq 4$$

Step 2: Write the solution

The solution to the inequality is $x \geq 4$.

Example 2

Solve the inequality $\log_3 y \leq 1$ for the variable $y$.

Step 1: Rewrite the inequality in exponential form

When we rewrite the inequality in exponential form, we get:

$$y \leq 3$$

Step 2: Write the solution

The solution to the inequality is $y \leq 3$.