Problem 4 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Elements -2, -1, 0, and \(\frac{1}{2}\) satisfy the inequality.

Step 1: Write the Inequality

The given inequality is \(x + 1 < 2\). To find the elements of the set \(S\) that satisfy this inequality, subtract 1 from both sides to isolate \(x\).

Step 2: Simplify the Inequality

Simplify the inequality \(x + 1 < 2\) by subtracting 1 from both sides, resulting in \(x < 1\).

Step 3: Analyze Each Element of S

Examine each element of the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) to determine which satisfy \(x < 1\).

Step 4: Check Element -2

For \(x = -2\), \(-2 < 1\) is true, so \(-2\) satisfies the inequality.

Step 5: Check Element -1

For \(x = -1\), \(-1 < 1\) is true, so \(-1\) satisfies the inequality.

Step 6: Check Element 0

For \(x = 0\), \(0 < 1\) is true, so \(0\) satisfies the inequality.

Step 7: Check Element \(\frac{1}{2}\)

For \(x = \frac{1}{2}\), \(\frac{1}{2} < 1\) is true, so \(\frac{1}{2}\) satisfies the inequality.

Step 8: Check Element 1

For \(x = 1\), \(1 < 1\) is false, so 1 does not satisfy the inequality.

Step 9: Check Element \(\sqrt{2}\)

For \(x = \sqrt{2}\), since \(\sqrt{2} \approx 1.414\), \(\sqrt{2} < 1\) is false, so \(\sqrt{2}\) does not satisfy the inequality.

Step 10: Check Element 2

For \(x = 2\), \(2 < 1\) is false, so 2 does not satisfy the inequality.

Step 11: Check Element 4

For \(x = 4\), \(4 < 1\) is false, so 4 does not satisfy the inequality.

Step 12: List Satisfying Elements

The elements of \(S\) that satisfy the inequality \(x < 1\) are \(-2, -1, 0, \frac{1}{2}\).

Key Concepts

Solving Inequalities

Solving inequalities is quite similar to solving standard equations but has some unique traits. Here we focus on the inequality given: \(x + 1 < 2\). The goal is to find values of \(x\) that make the inequality true.
In the process:

• First, simplify the expression. Here, subtract 1 from both sides, providing \(x < 1\). This marks our solution condition.
• In inequalities, remember flipping the sign when multiplying or dividing by a negative number, but that's not needed here.
• Check each value of the set \(S\) against \(x < 1\), to identify valid solutions.

Understanding how to handle inequalities is a vital skill. It helps solve real-world problems, where sometimes all we need is a range of possibilities instead of exact values.

Set Notation

Set notation is a way to list numbers or objects fulfilling specific conditions. In this exercise, we use it to determine which numbers in set \(S\) satisfy the inequality.
The given set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\). We need to check each element of this set to see if it meets our condition \(x < 1\).

• Analyzing, \(-2, -1, 0, \frac{1}{2}\) all satisfy \(x < 1\).
• The other numbers, \(1, \sqrt{2}, 2, 4\), do not.

This shows how set notation helps in organizing a group of solutions, simplifying results, and representing them in a neat and understandable format.

Algebraic Expressions

Algebraic expressions involve variables, numbers, and operations. They serve as the foundation for inequalities and equations like the one in our example: \(x + 1 < 2\).

• Expressions allow us to represent relationships mathematically. They help us make logical inferences, reducing or transforming expressions where needed.
• In this problem, add and subtract operations involve isolating \(x\).
• Evaluating each element from set \(S\) against this algebraic expression establishes which values are valid, showcasing a practical algebraic application.

Being fluent in handling algebraic expressions is crucial. They provide the basis for tackling more complex mathematical problems and finding diverse solutions.