Problem 6 - Algebra and Trigonometry

Question

\(1-8=\) Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ -2 \leq 3-x<2 $$

Step-by-Step Solution

Answer

Elements \(\sqrt{2}, 2,\) and \(4\) satisfy the inequality.

Step 1: Understanding the Inequality

The inequality given is \[-2 \leq 3-x < 2\]This is a compound inequality, meaning it must satisfy two conditions simultaneously: \( -2 \leq 3-x \) and \( 3-x < 2 \).

Step 2: Solve the First Inequality

First, solve the inequality \[-2 \leq 3 - x\] Rearrange the inequality: Add \(x\) to both sides: \[-2 + x \leq 3\] Then subtract 3 from both sides: \[x \leq 5\]

Step 3: Solve the Second Inequality

Now, solve the inequality \[3 - x < 2\] Subtract 3 from both sides: \[-x < -1\] Then multiply both sides by -1 (remember this reverses the inequality): \[x > 1\]

Step 4: Combine the Solutions

Combine the results from both inequalities:The solution must satisfy both conditions, \(x \leq 5\) and \(x > 1\). Therefore, the combined inequality is \[1 < x \leq 5\].

Step 5: Test Elements from Set S

Check each element in the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) to see if they satisfy \(1 < x \leq 5\):- \(-2, -1, 0, \frac{1}{2}, 1\) do not satisfy \(1 < x\).- Check \(\sqrt{2}\): approximately \(1.414\), which is greater than 1.- \(2\) satisfies \(1 < 2 \leq 5\).- \(4\) satisfies \(1 < 4 \leq 5\).Thus, the elements that satisfy \(1 < x \leq 5\) are \(\sqrt{2}\), \(2\), and \(4\).

Key Concepts

Compound Inequalities

Compound inequalities include two separate inequalities that must both be true at the same time. Think of them as two conditions that must be met simultaneously.
In the exercise, we are dealing with \[-2 \leq 3-x < 2.\] This is a compound inequality.

• The first part, \[-2 \leq 3-x,\] implies that \(x\) shifted positions by adding \(x\) establishes a threshold on the other side.
• Similarly, the second part, \[3-x < 2,\] informs us that subtracting 3 sets another limit.

The trick is to find a range for \(x\) where both inequalities hold true. Imagine drawing the two inequalities on a number line and analyzing where they overlap. This helps to visualize the valid range for all elements of the set.

Solving Inequalities

Solving inequalities can first be attempted similarly to equations, but take care to remember that the direction of an inequality symbol can change. This happens when you multiply or divide by a negative number.
Let's see how each inequality was addressed in the steps:

• For \[-2 \leq 3-x,\] the inequality was rearranged:
  • Add \(x\) to both sides: \[-2 + x \leq 3,\]
  • Then subtract 3 to isolate \(x\): \[x \leq 5.\]
• For \[3-x < 2,\] managing negatives was crucial:
  • Subtracting 3: \[-x < -1,\]
  • And then multiplying by -1, reversing the sign: \[x > 1.\]

In summary, the solutions from these inequalities defined a common range, \[1 < x \leq 5.\] This means that for a value of \(x\) to make both inequalities true, it has to be greater than 1 but no more than 5.

Set Notation

Set notation is a neat way to display the subsets of values a variable can have. In this exercise, we use it to showcase which elements from set \(S\) fit the criteria defined by inequalities.
The set given is \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}.\) Our task was to find which elements of this set satisfy\[1 < x \leq 5.\]

• Values like \(-2, -1, 0, \frac{1}{2}, 1\) did not satisfy because they didn't meet the lower threshold, which is more than 1.
• Values such as \(\sqrt{2}\), approximately 1.414, do fit the criteria since they're greater than 1.
• Additionally, \(2\) and \(4\) also satisfy \(1 < x \leq 5.\)

The subset that meets the required condition using set notation is \(\{\sqrt{2}, 2, 4\}.\) Using set notation efficiently summarized which specific elements within the given set satisfied the compound inequality.