Solve The Inequality For $v$.$-\frac{9}{2}v \ \textless \ 6$Simplify Your Answer As Much As Possible.$\square$

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**Solving Inequalities: A Step-by-Step Guide to Understanding $-\frac{9}{2}v \ \textless \ 6$**

What is an Inequality?

An inequality is a mathematical statement that compares two values or expressions using a combination of numbers, variables, and mathematical operations. Inequalities are used to describe relationships between quantities that are not equal. In this article, we will focus on solving the inequality −92v \textless 6-\frac{9}{2}v \ \textless \ 6.

Understanding the Inequality

The given inequality is −92v \textless 6-\frac{9}{2}v \ \textless \ 6. To solve this inequality, we need to isolate the variable vv on one side of the inequality sign. The inequality sign $\ \textless \ $ indicates that the value of vv is less than the value on the right-hand side of the inequality.

Step 1: Multiply Both Sides by -1/2

To isolate the variable vv, we need to get rid of the coefficient −92-\frac{9}{2} that is being multiplied by vv. We can do this by multiplying both sides of the inequality by the reciprocal of the coefficient, which is −29-\frac{2}{9}.

-\frac{9}{2}v \  \textless \  6
\Rightarrow \  v \  \textless \  -\frac{2}{9} \times 6
\Rightarrow \  v \  \textless \  -\frac{12}{9}
\Rightarrow \  v \  \textless \  -\frac{4}{3}

Step 2: Simplify the Right-Hand Side

The right-hand side of the inequality is −43-\frac{4}{3}. This is a fraction, and we can simplify it by dividing the numerator and denominator by their greatest common divisor, which is 1.

v \  \textless \  -\frac{4}{3}

Step 3: Write the Solution in Interval Notation

The solution to the inequality is all values of vv that are less than −43-\frac{4}{3}. We can write this in interval notation as (−∞,−43)(-\infty, -\frac{4}{3}).

(-\infty, -\frac{4}{3})

Frequently Asked Questions

Q: What is the meaning of the inequality sign $\ \textless \ $?

A: The inequality sign $\ \textless \ $ indicates that the value of vv is less than the value on the right-hand side of the inequality.

Q: How do I solve an inequality with a fraction coefficient?

A: To solve an inequality with a fraction coefficient, you need to multiply both sides of the inequality by the reciprocal of the coefficient.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality as a range of values. It is denoted by a pair of parentheses or brackets, with the lower bound on the left and the upper bound on the right.

Q: How do I determine the direction of the inequality sign?

A: The direction of the inequality sign depends on the sign of the coefficient. If the coefficient is positive, the inequality sign is $\ \textless \ $. If the coefficient is negative, the inequality sign is $\ \textgreater \ $.

Conclusion

Solving inequalities is an important skill in mathematics, and it requires a clear understanding of the concepts and techniques involved. In this article, we have solved the inequality −92v \textless 6-\frac{9}{2}v \ \textless \ 6 and provided a step-by-step guide to understanding the solution. We have also answered some frequently asked questions to help you better understand the concepts involved.