Problem 37 - Calculus Single Variable

Question

Sketch the set on a real number line. \(\\{y: y-\sqrt{7}<3 y+4 \leq 4 y+\sqrt{2}\\}\)

Step-by-Step Solution

Answer

The solution is all numbers \( y \) such that \( y \geq 4 - \sqrt{2} \).

Step 1: Clarify the set notation

We start by analyzing the set notation given: \(\{y: y - \sqrt{7} < 3y + 4 \leq 4y + \sqrt{2}\}\). This means we have two inequalities combined: \( y - \sqrt{7} < 3y + 4 \) and \( 3y + 4 \leq 4y + \sqrt{2} \). We will solve these inequalities separately.

Step 2: Solve the first inequality

The first inequality is \( y - \sqrt{7} < 3y + 4 \). Rearrange the terms: \( y - 3y < 4 + \sqrt{7} \). This further simplifies to \( -2y < 4 + \sqrt{7} \). Divide by -2 (remember to flip the inequality sign): \( y > -2 - \frac{\sqrt{7}}{2} \).

Step 3: Solve the second inequality

The second inequality is \( 3y + 4 \leq 4y + \sqrt{2} \). Rearrange the terms: \( 3y + 4 - 4y \leq \sqrt{2} \). This simplifies to \( -y \leq \sqrt{2} - 4 \). Divide by -1 (remember to flip the inequality sign): \( y \geq 4 - \sqrt{2} \).

Step 4: Find the intersection of the solutions

Combine the results from the two inequalities to find a common solution: \( y > -2 - \frac{\sqrt{7}}{2} \) and \( y \geq 4 - \sqrt{2} \). The intersection of these conditions is \( y \geq 4 - \sqrt{2} \) since \( 4 - \sqrt{2} > -2 - \frac{\sqrt{7}}{2} \).

Step 5: Sketch the set on the number line

To sketch the solution \( y \geq 4 - \sqrt{2} \) on a real number line, draw an open circle at \( 4 - \sqrt{2} \) and shade the line extending to the right, including \( 4 - \sqrt{2} \). This represents all numbers \( y \) satisfying the inequality.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that relate two values, showing if one is greater or lesser than the other. When solving inequalities, the goal is to find all possible values of the variable that make the inequality true. In our exercise, we dealt with two inequalities:

• \( y - \sqrt{7} < 3y + 4 \)
• \( 3y + 4 \leq 4y + \sqrt{2} \)

To solve these, you must rearrange terms to isolate the variable on one side while ensuring any arithmetic manipulation respects inequality rules. Specifically, when multiplying or dividing both sides of an inequality by a negative number, it is crucial to reverse the direction of the inequality symbol.
For instance, after simplifying the first inequality, we rearrange to achieve the inequality \( -2y < 4 + \sqrt{7} \) and divide by \(-2\), resulting in \( y > -2 - \frac{\sqrt{7}}{2} \), flipping the inequality sign in the process.

Intersection of Solutions

Once we solve each inequality, the next step is to find where their solutions overlap—this is known as finding the intersection of solutions. The solution to the intersection is the set of values that satisfy all given inequalities simultaneously. In our exercise, we determined:

• From the first inequality, \( y > -2 - \frac{\sqrt{7}}{2} \)
• From the second inequality, \( y \geq 4 - \sqrt{2} \)

For these inequalities, we observe that the intersection is \( y \geq 4 - \sqrt{2} \). This means the set of solutions that meet both criteria starts from \( 4 - \sqrt{2} \) and goes upwards. We find this intersection by acknowledging that \(4 - \sqrt{2}\) is larger than \(-2 - \frac{\sqrt{7}}{2}\), so the solutions start from \( 4 - \sqrt{2} \). Only values greater than or equal to this point satisfy both conditions.

Real Number Line Representation

A real number line is a visual tool used to represent inequalities and solutions. When you plot a solution on a number line, it becomes easier to understand the range of possible values. For the solution \( y \geq 4 - \sqrt{2} \), you would:

• Locate the point \(4 - \sqrt{2}\) on the number line.
• Place a filled-in circle at \(4 - \sqrt{2}\) to indicate that this value is included in the solution.
• Shade or draw an arrow to the right from this point to reflect all numbers greater than \(4 - \sqrt{2}\).

This shaded region signifies all potential values \(y\) can take to make the inequality true. Using a real number line helps you quickly see the range and nature of solutions, making inequalities more tangible.