Problem 8 - 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. $$-2 \leq 3-x<2$$

Step-by-Step Solution

Answer

The elements that satisfy the inequality are \(\sqrt{2}\), \(2\), and \(4\).

Step 1: Translate the Inequality

The inequality given is \(-2 \leq 3-x < 2\). This can be interpreted as a double inequality that must be satisfied simultaneously: \(-2 \leq 3-x\) and \(3-x < 2\).

Step 2: Solve the First Part of the Inequality

Start with the inequality \(-2 \leq 3-x\). Subtract 3 from both sides to isolate \(-x\), which gives \(-5 \leq -x\). Multiply both sides by \(-1\) and reverse the inequality sign to get \(x \leq 5\).

Step 3: Solve the Second Part of the Inequality

Now, solve the second part of the inequality: \(3-x < 2\). Subtract 3 from both sides to obtain \(-x < -1\). Multiplying both sides by \(-1\) and reversing the inequality sign gives \(x > 1\).

Step 4: Combine the Results

Combine the results from the steps above to form a compound inequality: \(1 < x \leq 5\).

Step 5: Identify the Elements of Satisfying the Inequality

Check each element of the set \(S=\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) to see which satisfies the inequality \(1 < x \leq 5\).

Step 6: Verify Each Element

- For \(-2, -1, 0, \frac{1}{2}, 1\): These values do not satisfy \(x > 1\).- For \(\sqrt{2}\): It is approximately 1.414, satisfying \(1 < x \leq 5\).- For \(2\), \(4\): These satisfy both \(1 < x\) and \(x \leq 5\).

Step 7: List the Satisfying Elements

The elements of \(S\) that satisfy \(1 < x \leq 5\) are \(\sqrt{2}\), \(2\), and \(4\).

Key Concepts

Double Inequalities

Double inequalities are a way of expressing two inequalities at once. They involve mathematical expressions that include two inequality signs. For example, the inequality \(-2 \leq 3 - x < 2\) is a double inequality.
This means that \(-2\) is less than or equal to \(3-x\) and at the same time, \(3-x\) is less than \(2\). Both conditions need to be true simultaneously for the entire expression to hold.
Working with double inequalities typically involves breaking them into two separate but related parts:

• Solving each part separately.
• Combining the solutions to find a common range where both are true.

It is crucial to remember that when manipulating inequalities, such as multiplying or dividing by a negative number, the inequality sign must be flipped. This is because multiplying or dividing by a negative reverses the order of numbers on a number line.

Compound Inequalities

Compound inequalities are created when two inequalities are joined together. In the case of double inequalities, both inequalities must be true at the same time for the compound inequality to be satisfied.
For example, after solving the initial double inequalities in the problem, \(-2 \leq 3-x\) and \(3-x < 2\), we derive the compound inequality \(1 < x \leq 5\).
This statement tells us that the variable \(x\) must be greater than \(1\) but also less than or equal to \(5\). When solving such compound inequalities, we:

• First find the solution set for each individual inequality.
• Then determine the intersection of these sets, which provides the solution for the compound inequality.

The intersection of the solution sets includes all numbers that satisfy both individual inequalities, resulting in a range of values for which the compound inequality holds true.

Set Elements Evaluation

Set elements evaluation is the process of checking which elements from a given set satisfy an inequality or condition. In this exercise, we have both a compound inequality \(1 < x \leq 5\) and a set \(S=\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\).
To evaluate, we:

• Check each element of the set \(S\) against the compound inequality.
• Determine which elements make the inequality true.

For this particular exercise, elements like \(\sqrt{2}\), \(2\), and \(4\) satisfy the compound inequality \(1 < x \leq 5\), because:

• \(\sqrt{2}\) approximately equals \(1.414\) which is within the interval.
• Both \(2\) and \(4\) clearly fall between \(1\) and \(5\).

This method ensures that only elements that meet the criteria of the inequality are selected from the set. It involves inspecting each element, which can often be a straightforward verification process, especially with smaller sets.