Problem 33 - Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. See Example 4. $$ \sqrt{5-x}+10=9 $$

Step-by-Step Solution

Answer

No solutions exist, as the square root cannot be negative.

Step 1: Isolate the Square Root

Start by isolating the square root expression on one side of the equation. Given the equation is \( \sqrt{5-x} + 10 = 9 \), first move 10 to the other side by subtracting it from both sides of the equation.\[ \sqrt{5-x} = 9 - 10 \]\[ \sqrt{5-x} = -1 \]

Step 2: Determine the Possibility of the Solution

Recognize that a square root cannot equal a negative number. Since the square root of a number results in a non-negative value, the equation \( \sqrt{5-x} = -1 \) has no real solutions.

Step 3: Analyze for Extraneous Solutions

Since no real number can satisfy \( \sqrt{5-x} = -1 \), we determine there are no solutions to this equation. Therefore, any 'solutions' that might appear during this process would be extraneous by their nature.

Key Concepts

Square Root

The concept of a square root is fundamental in algebra and helps in solving equations involving quadratic expressions. A square root of a number \( a \) is essentially a number \( x \) such that \( x^2 = a \). In simpler terms, when we square \( x \), we get \( a \).

For instance, the square root of 9 is 3 because \( 3^2 = 9 \). It's important to understand that square roots typically refer to the principal (or non-negative) root. Hence the square root function \( \sqrt{x} \) always gives a non-negative number.

• This means for any number \( x \), \( \sqrt{x} \geq 0 \).
• In equations, if you find \( \sqrt{a} = -1 \), this indicates a fundamental issue because \( \sqrt{a} \) must always be non-negative.

Recognizing this allows us to immediately identify when a problem has no real solutions, as seen in the given exercise where \( \sqrt{5-x} \) cannot be \( -1 \).

Understanding square roots is essential for identifying errors or simplifications in solving equations.

Extraneous Solutions

Extraneous solutions are values that emerge as solutions in the process of solving an equation but do not satisfy the original equation. They often appear when both sides of an equation are squared.

Here's why extraneous solutions occur:

• When you square both sides of an equation, it might introduce extra solutions because squaring "neutralizes" negative numbers.
• The original equation was transformed, which can lead to results that don't apply to the initial setup.

In the case of the given exercise, no squaring was done. However, if there had been solutions to this equation that did not satisfy the condition after substitution, they would be classified as extraneous.

It's crucial to double-check solutions by substituting them back into the original equation to ensure they're valid.

Real Numbers

Real numbers are a central concept in mathematics, comprising all the numbers that can be found on the number line. This includes both rational numbers like 2, -5, and 4.5; and irrational numbers like \( \pi \) and \( \sqrt{2} \).

Real numbers have several key attributes:

• They can be positive, negative, or zero.
• They can be whole numbers, fractions, or decimals.
• They provide the foundation for discussing number properties and behavior in equations.

In the exercise at hand, understanding that a square root cannot be negative crucially tells us why \( \sqrt{5-x} = -1 \) holds no real solutions.

Knowing the properties of real numbers helps in recognizing when an equation has no real solutions and in identifying equations that require no further solving due to their inherent impossibility.