Solve The Inequality For V V V . − 7 8 ≥ V + 4 -\frac{7}{8} \geq V + 4 − 8 7 ≥ V + 4 Simplify Your Answer As Much As Possible.

Mar 10, 2025 by ADMIN 129 views

**Solving Inequalities: A Step-by-Step Guide**

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $-\frac{7}{8} \geq v + 4$ and simplify our answer as much as possible.

Understanding the Inequality

The given inequality is $-\frac{7}{8} \geq v + 4$ . To solve this inequality, we need to isolate the variable $v$ on one side of the inequality sign. The first step is to subtract 4 from both sides of the inequality.

Subtracting 4 from Both Sides

When we subtract 4 from both sides of the inequality, we get:

-\frac{7}{8} - 4 \geq v + 4 - 4

Simplifying the left-hand side, we get:

-\frac{7}{8} - \frac{32}{8} \geq v

Combining the fractions on the left-hand side, we get:

-\frac{39}{8} \geq v

Understanding the Direction of the Inequality

The direction of the inequality is important to note. Since we subtracted 4 from both sides, the direction of the inequality remains the same. If the original inequality was $-\frac{7}{8} \geq v + 4$ , then the direction of the inequality is still greater than or equal to (≥).

Isolating the Variable

Now that we have isolated the variable $v$ on one side of the inequality, we can write the solution as:

v \leq -\frac{39}{8}

Simplifying the Answer

To simplify the answer, we can convert the fraction to a decimal or a mixed number. Converting the fraction to a decimal, we get:

v \leq -4.875

Converting the fraction to a mixed number, we get:

v \leq -4\frac{7}{8}

Conclusion

In conclusion, the solution to the inequality $-\frac{7}{8} \geq v + 4$ is $v \leq -\frac{39}{8}$ . We can simplify the answer by converting the fraction to a decimal or a mixed number. The direction of the inequality is still greater than or equal to (≥).

Tips and Tricks

When solving inequalities, it's essential to maintain the direction of the inequality.
When subtracting or adding a value to both sides of the inequality, the direction of the inequality remains the same.
When multiplying or dividing both sides of the inequality by a negative value, the direction of the inequality is reversed.

Real-World Applications

Solving inequalities has numerous real-world applications. For example, in finance, inequalities can be used to determine the minimum or maximum value of an investment. In engineering, inequalities can be used to determine the minimum or maximum value of a physical quantity, such as temperature or pressure.

Common Mistakes

When solving inequalities, it's common to forget to maintain the direction of the inequality.
When subtracting or adding a value to both sides of the inequality, it's common to forget to change the direction of the inequality.
When multiplying or dividing both sides of the inequality by a negative value, it's common to forget to reverse the direction of the inequality.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities and provided a step-by-step guide on how to isolate the variable on one side of the inequality sign. In this article, we will answer some frequently asked questions about solving inequalities.

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

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Factor the quadratic expression, if possible.
Set each factor equal to zero and solve for $x$ .
Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a rational inequality and a polynomial inequality?

A: A rational inequality is an inequality that can be written in the form $\frac{f(x)}{g(x)} \geq 0$ or $\frac{f(x)}{g(x)} \leq 0$ , where $f(x)$ and $g(x)$ are polynomials and $x$ is the variable. A polynomial inequality, on the other hand, is an inequality that can be written in the form $f(x) \geq 0$ or $f(x) \leq 0$ , where $f(x)$ is a polynomial and $x$ is the variable.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

Factor the numerator and denominator, if possible.
Set each factor equal to zero and solve for $x$ .
Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$ , where $a$ , $b$ , and $c$ are constants and $x$ is the variable. A nonlinear inequality, on the other hand, is an inequality that cannot be written in the form $ax + b \geq c$ or $ax + b \leq c$ , where $a$ , $b$ , and $c$ are constants and $x$ is the variable.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you can use the following steps:

Use a graph or a number line to determine the intervals where the inequality is true.
Use algebraic methods, such as factoring or the quadratic formula, to solve the inequality.

Q: What is the difference between a one-sided inequality and a two-sided inequality?

A: A one-sided inequality is an inequality that can be written in the form $ax + b \geq c$ or $ax + b \leq c$ , where $a$ , $b$ , and $c$ are constants and $x$ is the variable. A two-sided inequality, on the other hand, is an inequality that can be written in the form $ax + b \geq c$ and $ax + b \leq d$ , where $a$ , $b$ , $c$ , and $d$ are constants and $x$ is the variable.

Q: How do I solve a two-sided inequality?

A: To solve a two-sided inequality, you can use the following steps:

Solve each one-sided inequality separately.
Use the solution to each one-sided inequality to determine the intervals where the two-sided inequality is true.

Conclusion

Solving inequalities is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with ease. Remember to maintain the direction of the inequality, and don't forget to change the direction of the inequality when multiplying or dividing both sides by a negative value. With practice and patience, you can become proficient in solving inequalities and apply them to real-world problems.

Tips and Tricks

When solving inequalities, it's essential to maintain the direction of the inequality.
When subtracting or adding a value to both sides of the inequality, the direction of the inequality remains the same.
When multiplying or dividing both sides of the inequality by a negative value, the direction of the inequality is reversed.
When solving a quadratic inequality, you can use the quadratic formula to find the solutions.
When solving a rational inequality, you can use the factoring method to find the solutions.

Real-World Applications

Common Mistakes

When solving inequalities, it's common to forget to maintain the direction of the inequality.
When subtracting or adding a value to both sides of the inequality, it's common to forget to change the direction of the inequality.
When multiplying or dividing both sides of the inequality by a negative value, it's common to forget to reverse the direction of the inequality.
When solving a quadratic inequality, it's common to forget to use the quadratic formula.
When solving a rational inequality, it's common to forget to factor the numerator and denominator.