Problem 27 - Precalculus

Question

Solve the equation or inequality. $$10-\sqrt{x-2} \leq 11$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 2\).

Step 1: Isolate the square root term

To solve the inequality, isolate the square root on one side of the inequality. Start with the given inequality: \[10 - \sqrt{x-2} \leq 11\]Subtract 10 from both sides of the inequality to get:\[-\sqrt{x-2} \leq 1\]

Step 2: Remove the negative sign

To remove the negative sign in front of the square root, multiply both sides of the inequality by -1. Remember to reverse the inequality sign when you multiply by a negative:\[\sqrt{x-2} \geq -1\]

Step 3: Analyze the square root properties

The square root of a real number is always non-negative. Therefore, we must ensure that:\[\sqrt{x-2} \geq 0\]Combine this with the previous result, and we have:\[\sqrt{x-2} \geq 0\]

Step 4: Remove the square root

Since the square root is greater than or equal to 0, solve for \(x-2 \geq 0\) to remove the square root:\[x - 2 \geq 0\]

Step 5: Solve for x

Add 2 to both sides to solve for \(x\):\[x \geq 2\]

Step 6: Combine with square root condition

Considering back to Step 2 that \(\sqrt{x-2} \geq -1\) is inherently true because \(\sqrt{x-2}\) cannot be negative, the constraints show that initially \(x \geq 2\) is our only requirement.

Key Concepts

Square Root

A square root is essentially the operation to determine a number which, when multiplied by itself, yields the original number. For example, the square root of 4 is 2 because 2 multiplied by 2 equals 4. Square roots are often represented with the symbol \( \sqrt{} \). Understanding the properties of square roots is crucial, especially since they always result in non-negative numbers (when dealing with real numbers).

When solving inequalities or equations that involve square roots, it's important to remember:

• The expression under the square root, also known as the radicand, must be non-negative. For example, in \( \sqrt{x-2} \), \( x-2 \geq 0 \).
• Square roots produce results that are equal to or greater than zero.
• Manipulating square roots in equations involves understanding their limitations in the solution set.

Using these principles helps guide solving inequalities like \( 10 - \sqrt{x-2} \leq 11 \), ensuring that all steps respect the nature of square roots.

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules to manipulate these symbols. It provides a means for solving problems and finding unknown values, represented typically by symbols like \( x \) or \( y \).

When tackling algebraic inequalities, like in this exercise, the objective is to isolate the variable on one side. This requires a series of operations:

• Isolating Terms: Subtract or add terms from both sides to begin isolating the variable. For example, subtract 10 from both sides in \( 10 - \sqrt{x-2} \leq 11 \) yields \( -\sqrt{x-2} \leq 1 \).
• Handling Negative Signs: Multiplying or dividing by negative numbers involves flipping the inequality sign. In the inequality \( -\sqrt{x-2} \leq 1 \), multiplying by \(-1\) gives \( \sqrt{x-2} \geq 0 \).
• Combining Conditions: Ensure all conditions from manipulation like square root properties are met before final steps.

In algebra, each of these steps follows logically to simplify and ultimately solve an equation or inequality.

Real Numbers

Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers. Rational numbers are those you can express as a fraction, like \( \frac{1}{2} \) or \( 3 \). Irrational numbers are those that cannot be expressed as fractions, like \( \sqrt{2} \) or \( \pi \).

Real numbers are crucial in algebra, especially when determining the domain of functions or solutions to equations. In the context of square roots, it’s critical to understand:

• All square root expressions must have non-negative radicands for real solutions. This means, if \( x-2 \) is under a square root sign, then \( x-2 \) must be greater than or equal to zero.
• Real solutions constrain the possible set of \( x \) values to those that do not result in imaginary numbers (i.e., square roots of negative numbers).
• If the inequality is true for all real numbers, as in the case after Step 3, it further simplifies the problem.

Recognizing the properties of real numbers helps ensure that solutions are valid and complete in solving problems like these.