Problem 10 - Precalculus Mathematics for Calculus

Question

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$x^{2}+2<4$$

Step-by-Step Solution

Answer

The elements satisfying the inequality are \(-1, 0, \frac{1}{2}, \) and \( 1\).

Step 1: Rewrite the Inequality

Start by rewriting the inequality as follows: If we have the inequality \(x^2 + 2 < 4\), we can simplify it by subtracting 2 from both sides: \(x^2 < 2\). This is a more straightforward form of the inequality that will help us determine which elements satisfy it.

Step 2: Substitute Elements into the Inequality

Next, we need to check each element in the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) to determine if they satisfy \(x^2 < 2\):- For \(x = -2\), \((-2)^2 = 4\); does not satisfy.- For \(x = -1\), \((-1)^2 = 1\); satisfies.- For \(x = 0\), \(0^2 = 0\); satisfies.- For \(x = \frac{1}{2}\), \((\frac{1}{2})^2 = \frac{1}{4}\); satisfies.- For \(x = 1\), \(1^2 = 1\); satisfies.- For \(x = \sqrt{2}\), \((\sqrt{2})^2 = 2\); does not satisfy.- For \(x = 2\), \(2^2 = 4\); does not satisfy.- For \(x = 4\), \(4^2 = 16\); does not satisfy.

Step 3: List the Satisfying Elements

Based on the calculations from Step 2, the elements that satisfy the inequality \(x^2 < 2\) are \(-1, 0, \frac{1}{2}, \) and \( 1\).

Key Concepts

Quadratic Inequality

Quadratic inequalities are mathematical expressions involving quadratic polynomials that use inequality signs like ">", "<", "≥", or "≤". They form a core part of algebra and help solve real-world problems where a range of values is needed. For our exercise, the quadratic inequality is originally given as:\[x^{2} + 2 < 4.\]After a few manipulation steps, it becomes simpler to solve by taking the form:\[x^{2} < 2.\]This is considered a quadratic inequality because it involves the square of variable \(x\) and aims to find values of \(x\) that satisfy the condition. Understanding quadratic inequalities involves determining the critical points where related equations are equal, dividing the number line, and then testing these intervals to find true regions for the inequality.

Set of Numbers

A set is a collection of distinct objects or numbers. In this case, we have a set \( S \) given by:\[ S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \].In the context of solving inequalities, we need to determine which numbers in the set satisfy the given inequality condition. Not all numbers in a set will fulfill a given inequality. Our task involves evaluating each number in \( S \) to see if it makes the inequality statement true.Understanding the properties of each number—be it negative, positive, a fraction, or involving a square root—helps in this evaluation process. Assessing each element methodically ensures that we don't miss any potential solutions.

Substitution Method

The substitution method involves replacing variables in an expression or equation with specific values to check if they satisfy a condition. In our exercise:- We substitute each element of the set \( S\) into the simplified inequality \( x^2 < 2\).- By evaluating the inequality for each element:

• Calculate the square of the number.
• Compare the squared value to determine if it is less than 2.

For example, substituting \( x = -1 \):- Calculate \((-1)^2 = 1 \).- Since \(1 < 2\), \( x = -1 \) satisfies the inequality.This method allows us to systematically verify which elements satisfy the quadratic inequality without needing to graph or solve extensive equations.

Solving Inequalities

Solving inequalities means finding the set of values that make the inequality statement true. Here, solving means determining which elements of \( S \) satisfy the inequality \(x^2 < 2\).Steps to solve include:

• Simplifying the inequality to a more workable form, which we achieved with \( x^2 < 2 \).
• Applying the substitution method to test each element in the set \( S \).
• Selecting only those numbers where the calculated values meet the inequality condition.

In this instance, the satisfying elements are \(-1, 0, \frac{1}{2},\) and \(1\). This means these values conform to the condition of the inequality by yielding squares less than 2, showcasing mastery in identifying valid solutions to quadratic inequalities using substitution techniques.