Problem 29 - Beginning and Intermediate Algebra

Question

Solve each inequality. $$(3 v-11)^{2}>-20$$

Step-by-Step Solution

Answer

The inequality \((3v - 11)^2 > -20\) is true for all values of \(v\) since any squared expression is non-negative and always greater than a negative number. Therefore, the solution is \(v \in \mathbb{R}\).

Step 1: Observe the inequality

Notice that the left side of the inequality is a squared expression, which means it is non-negative. Therefore, it will always be greater than any negative number on the right side, such as -20.

Step 2: State the solution

Since any squared expression (including \((3v - 11)^2\)) is greater than any negative number (such as -20), the inequality is true for all values of \(v\). Thus, the solution is: $$v \in \mathbb{R}$$

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics that consist of variables, numbers, and operational symbols. For instance, in the exercise given, the expression \((3v - 11)^2\) is an algebraic expression. It includes:

• A variable, \(v\), which represents any number.
• Coefficients, which are the numbers multiplied by the variables—in this case, 3 is the coefficient of \(v\).
• Constants, such as 11, which are fixed values.

Algebraic expressions can be as simple as \(3v\) or more complex involving multiple terms and operations, including squares and polynomials. The key is to accurately identify each part, which then allows you to manipulate and solve the expression effectively.

Quadratic Inequalities

A quadratic inequality involves a quadratic expression, usually in a form like \(ax^2 + bx + c\), where the highest power of the variable is squared. In the exercise, the quadratic expression is \((3v - 11)^2\).

Here are a few things to keep in mind about quadratic inequalities:

• Just like quadratic equations, they may involve terms like \(x^2\).
• The solution can often be a range or set of values, rather than specific numbers.
• Quadratic inequalities are often solved by finding the roots of the corresponding quadratic equation and examining the intervals between these roots.
• However, in special cases like the given problem where the square of something is always non-negative, the inequality may simply be satisfied by all real numbers.

Understanding how to interpret and solve quadratic inequalities is essential for mathematics, as they appear frequently in both theoretical calculations and real-life applications.

Real Numbers

Real numbers (\(\mathbb{R}\)) include everything you might count or measure, from positive numbers like 1 and 2, through zero, to negative numbers like -1. In the context of the exercise, when we say \(v \in \mathbb{R}\), we mean that \(v\) can be any real number.

This encompasses all numbers that are not complex, including:

• Rational numbers, like fractions and whole numbers.
• Irrational numbers, such as \(\sqrt{2}\) or \(\pi\), that cannot be expressed as simple fractions.

Understanding real numbers is crucial because they form the foundation for most of the numbers we use in everyday tasks and scientific calculations. They provide us the number continuum needed for measuring and analyzing the world.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve mathematical problems. In the exercise, reasoning was used to conclude that since a squared expression is non-negative, it satisfies the inequality \((3v - 11)^2 > -20\) for all \(v\).

Here are the key components of strong mathematical reasoning:

• Observation - Note the properties of the expressions and inequalities such as non-negativity of squares.
• Logical Deduction - Apply logical principles to deduce facts, like knowing \((3v - 11)^2\) is always greater than a negative number.
• Generalization - Extract general conclusions from specific observations, as in concluding the given inequality holds for all real numbers \(v\).

Mathematical reasoning allows us to validate our solutions and ensures the conclusions are correct and applicable across different scenarios. It's not just about solving equations but understanding the underlying principles that govern these solutions.