Problem 16 - Elementary Algebra

Question

Write the proper restrictions that must be placed on the variable so that each expression represents a real number. $$ \sqrt{3 a+2} $$

Step-by-Step Solution

Answer

Answer: The restriction on the variable a is $$a \ge -\frac{2}{3}$$.

Step 1: Identify the expression inside the square root

The expression inside the square root is $$3a + 2$$.

Step 2: Determine the condition for the expression inside the square root to be non-negative

For the given expression to represent a real number, the expression inside the square root must be non-negative, i.e., $$3a + 2 \ge 0$$.

Step 3: Solve the inequality for a

To find the restrictions on a, we'll solve the inequality $$3a + 2 \ge 0$$. Subtract 2 from both sides of the inequality: $$3a \ge -2$$ Now, divide both sides by 3: $$a \ge -\frac{2}{3}$$

Step 4: State the restrictions on the variable a

The proper restriction on the variable a so that the expression $$\sqrt{3a+2}$$ represents a real number is $$a \ge -\frac{2}{3}$$.

Key Concepts

Real Numbers

Real numbers include all the numbers you can think of: whole numbers, fractions, decimals, and even those involving square roots. Real numbers appear all over math, so understanding them is important. Let's break it down a bit more:

• Natural numbers are your basic counting numbers: 1, 2, 3, and so on.
• Whole numbers include natural numbers and zero: 0, 1, 2, 3, etc.
• Integers go beyond whole numbers by adding in negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers are any numbers that can be written as a fraction, where both the numerator and the denominator are integers and the denominator isn't zero.
• Irrational numbers cannot be expressed accurately by a fraction, like \(rac{\pi}{2}\) or the square root of 2, \(\sqrt{2}\).

The expression \(\sqrt{3a+2}\) must give a number that falls within real numbers. This involves ensuring the value inside the square root is not negative, since the square root of a negative number is not a real number.

Inequalities

Inequalities are mathematical expressions used to compare values and establish a range in which those values can lie. These comparisons are done using symbols such as \(>\), \(<\), \(\ge\), and \(\le\), which represent 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to,' respectively.

For the expression \(\sqrt{3a + 2}\), we need the expression inside the square root, \(3a + 2\), to be non-negative to ensure it results in a real number outcome. This means we form the inequality \(3a + 2 \ge 0\).

Solving such inequalities involves similar steps to solving equations, but you need to pay attention to the inequality sign:

• First, treat the inequality like an equation to get the variable by itself on one side.
• Subtract or add the same amount from both sides of the inequality.
• Divide or multiply both sides by a positive number. Be cautious if it's a negative number, because the inequality sign flips direction!

In the case of \(3a \ge -2\), we divide by 3, resulting in \(a \ge -\frac{2}{3}\). This outcome tells us the restrictions on \(a\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding them is crucial because they form the building blocks of algebra. Here’s how they work:

• Variables, like \(a\) or \(x\), stand in for unknown numbers. They allow us to represent general relationships rather than specific numeric values.
• Operations include addition, subtraction, multiplication, and division, along with more complex operations like square roots.
• Expressions don't usually equate to anything unless they're part of an equation or inequality, like the cases \(3a + 2\) or \(3a + 2 \ge 0\).

In the exercise, the expression \(3a+2\) is under a square root, affecting how we deal with it. Solving or manipulating an expression always follows the rules set by algebra, ensuring we adhere to mathematical principles and confirm our solutions: a valid range for \(a\) that keeps \(\sqrt{3a + 2}\) as a real number means \(a\) must be equal to or greater than \(-\frac{2}{3}\). Each decision in manipulation ensures the expression stays valid and represents a real number.