Problem 22 - Calculus Single Variable

Question

Sketch the set on a real number line. \(\\{y: 7 y+4 \geq 2 y+1\\}\)

Step-by-Step Solution

Answer

The solution is \(y \geq -\frac{3}{5}\), shown as a closed dot at \(-\frac{3}{5}\) with shading to the right on a number line.

Step 1: Simplify the inequality

Start by simplifying the given inequality: \(7y + 4 \geq 2y + 1\). To do this, you need to get all terms involving \(y\) on one side and constant terms on the other side. Begin by subtracting \(2y\) from both sides, resulting in \(5y + 4 \geq 1\).

Step 2: Isolate the variable term

Now, subtract 4 from both sides of the inequality to isolate the term involving \(y\). This gives you \(5y \geq -3\).

Step 3: Solve for the variable

To solve for \(y\), divide both sides of the inequality by 5. This results in \(y \geq -\frac{3}{5}\). This solution tells you the range of values that \(y\) can take.

Step 4: Sketch the inequality on a number line

Plot the inequality \(y \geq -\frac{3}{5}\) on a real number line. Since the inequality is 'greater than or equal to,' you will use a closed dot at \(-\frac{3}{5}\) and shade the region to the right of the point to indicate all the values that \(y\) can take.

Key Concepts

Real Number Line

The real number line is a visual representation of all the real numbers in a continuous line. It helps us understand where numbers fall in relation to one another. On the number line:

• Each point corresponds to a real number.
• The center is typically denoted by 0, with positive numbers to the right and negative numbers to the left.
• This line extends infinitely in both directions, capturing all real numbers from negative to positive infinity.

When working with inequalities, the real number line becomes especially useful for depicting ranges or sets of numbers that satisfy these inequalities. For example, when we have an inequality like \(y \geq -\frac{3}{5}\), it tells us where on this line the values of y lie.

Solving Inequalities

Solving inequalities involves finding the set of values that satisfy the inequality. This process is similar to solving equations, but it includes some specific rules:

• You can add or subtract the same number from both sides without changing the inequality's direction.
• When you multiply or divide both sides by a positive number, the direction of the inequality remains unchanged.
• However, when you multiply or divide both sides by a negative number, the inequality direction reverses.

In our example inequality \(7y + 4 \geq 2y + 1\), the process begins by simplifying and isolating the variable. Solving it correctly ensures the right number set on the real number line.

Variable Isolation

Isolating the variable is a key step in solving inequalities, as it helps to simplify the expression and makes it easier to understand its implications:

• First, bring all terms involving the variable, like \(y\), to one side of the inequality.
• Then, move the constant terms to the opposite side.

In our inequality example, this was achieved by subtracting \(2y\) from both sides, resulting in \(5y + 4 \geq 1\), and then subtracting 4, leading to \(5y \geq -3\). This methodical approach clarifies the range of values that the variable \(y\) can take.

Plotting Inequalities

Plotting inequalities on a real number line graphically shows the solutions:

• For inequalities like \(y \geq -\frac{3}{5}\), place a closed dot at \(-\frac{3}{5}\), indicating that \(-\frac{3}{5}\) is included in the set.
• Shade the area to the right of this point to represent all y values greater than or equal to \(-\frac{3}{5}\).

The closed dot signifies that the boundary number is part of the solution set, thanks to the "equal to" part of the inequality. This visual approach nicely complements algebraic solutions, making it easier to understand which values of the variable satisfy the inequality.