Problem 64 - Elementary Algebra

Question

For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number. $$ \sqrt{-5 y+15} $$

Step-by-Step Solution

Answer

Answer: The expression √(-5y + 15) represents a real number for y ≤ 3.

Step 1: Identify the expression inside the square root

We are given the expression: $$ \sqrt{-5 y+15} $$ The expression inside the square root is -5y + 15.

Step 2: Determine the restrictions on y for a real number

The value inside the square root must be non-negative for the entire expression to be a real number. Therefore, we need to find the values of y that satisfy the inequality: $$ -5 y+15 \geq 0 $$

Step 3: Solve the inequality

To solve the inequality -5y + 15 ≥ 0, we follow these steps: 1. Add 5y to both sides: $$ 15 \geq 5y $$ 2. Divide both sides by 5: $$ 3 \geq y $$

Step 4: Write the restrictions as an interval

In interval notation, the proper restriction for y to ensure that the given expression represents a real number is: $$ y \leq 3 $$ This means that the given expression will be a real number for any value of y less than or equal to 3.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that compare two values and show the relationship between them. Unlike equations, which state that two quantities are equal, inequalities indicate that one quantity is greater than, less than, or possibly equal to another. When solving inequalities, the goal is to find all possible values that make the inequality true.

In the given exercise, the inequality to solve is \( -5y + 15 \geq 0 \). To solve this, we must isolate the variable \(y\) on one side of the inequality. Subtracting 15 from both sides, we get \( -5y \geq -15 \). Next, to get \(y\) by itself, we divide both sides by -5, remembering to reverse the inequality sign because we're dividing by a negative number. This gives us \( y \leq 3 \).

Key steps when solving inequalities include:

• Maintaining the inequality balance by performing the same operations on both sides.
• Reversing the inequality sign when multiplying or dividing by a negative number.
• Graphing the solutions on a number line or using interval notation to express the solution set.

Solving inequalities is a fundamental skill in algebra that allows us to define the domain of variable quantities in various mathematical contexts.

Square Root Properties

The square root properties are based on the fundamental definition of a square root. For any non-negative real number \(x\), the square root of \(x\), denoted as \(\sqrt{x}\), is a number that, when squared, equals \(x\). An essential property is that the square root of a negative number is not a real number, but an imaginary one.

Therefore, to ensure an expression under a square root sign represents a real number, the value beneath the square root must be non-negative. This is the reason behind the restriction in the exercise \( \sqrt{-5 y+15} \), where \( -5y + 15 \geq 0 \). Other important properties of square roots include:

• The product property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), for non-negative \(a\) and \(b\).
• The quotient property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), provided \(b\) is not zero and \(a\) and \(b\) are non-negative.
• That \( \sqrt{a^2} = |a| \), the absolute value of \(a\), because the square root itself always gives a non-negative result.

These properties are the backbone for solving equations and inequalities involving square roots and for finding the domains of various functions.

Interval Notation

In mathematics, interval notation is a way of representing a set of numbers between two endpoints. It's often used to describe the solution sets of inequalities because it provides a concise way to express ranges of continuous numbers.

For example, the solution to our inequality \( y \leq 3 \) is best described in interval notation as \( (-\infty, 3] \) where:

• \( (-\infty, 3] \) indicates all numbers up to and including 3.
• The round parenthesis \( ( \) means that the endpoint is not included in the set (exclusive).
• The square bracket \( ] \) means that the endpoint is included in the set (inclusive).

Using interval notation can also avoid ambiguities that might arise when using inequalities alone. Other interval notation symbols include:\( for 'less than', \] or \)>em> for 'greater than', and their combinations to express inclusive or exclusive ranges. Interval notation is an efficient way to communicate solution sets and domains clearly and is a foundational concept in algebra and calculus.