Problem 69 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{1}{2} y+2 \geq \frac{1}{3} y-4 $$

Step-by-Step Solution

Answer

\( y \geq -36 \); solution in interval notation is \([-36, \infty)\).

Step 1: Eliminate the Fractions

To eliminate the fractions, first find the least common denominator (LCD) of the fractions involved. Here, the denominators are 2 and 3, so the LCD is 6. Multiply every term by 6 to clear the fractions:\[ 6 \left( \frac{1}{2}y + 2 \right) \geq 6 \left( \frac{1}{3}y - 4 \right) \]This simplifies to:\[ 3y + 12 \geq 2y - 24 \]

Step 2: Isolate the Variable Term

Subtract \(2y\) from both sides to isolate the variable term:\[ 3y - 2y + 12 \geq -24 \]Which simplifies to:\[ y + 12 \geq -24 \]

Step 3: Solve for the Variable

Subtract 12 from both sides to solve for \(y\):\[ y + 12 - 12 \geq -24 - 12 \]This simplifies to:\[ y \geq -36 \]

Step 4: Graph the Solution Set

On a number line, you would shade all numbers greater than or equal to \(-36\). This includes \(-36\) itself, represented by a closed circle or bracket.

Step 5: Write in Interval Notation

The solution set in interval notation is written as:\[ [-36, \infty) \]This indicates all numbers from \(-36\) to infinity, including \(-36\), but not infinity.

Key Concepts

Graphing Inequalities

When you solve inequalities, it's essential to understand the visual representation—how they look on a number line. Graphing inequalities involves recognizing three main components: the inequality sign, the boundary point, and how to represent this on a graph.

• **Inequality Sign:** Determines how you shade your graph. For example, \( \geq \) and \( \leq \) mean that the solution includes the boundary point, while \( > \) and \( < \) mean it doesn't.
• **Boundary Point:** In our exercise, the boundary point is \( -36 \). It is a crucial part of the solution and determines where to begin shading. Because our inequality is "\( y \geq -36\)", \( -36 \) is included.
• **Representation on the Number Line:** Use a filled dot or a closed bracket on \( -36 \) (since it is included). Shade everything to the right, because the inequality is greater than or equal to \( -36 \).

Graphing, therefore, confirms the solution visually and helps you see the range of possible solutions.

Interval Notation

Interval notation is a concise way to represent the set of solutions for an inequality. It uses brackets \((\) and \( ]\) to show whether endpoints are included, and parentheses \(()\) and \(\)) to indicate they are not.

In our case, the solution \( y \geq -36 \) translates to interval notation as \[ [-36, \infty) \]. Here's why:

• **Brackets \([, ]\):** Brackets mean that the endpoint is included in the interval. We use \([\) at \(-36\) because \( y \geq -36 \) means \(-36\) is part of the solution.
• **Parentheses \((, )\):** Parentheses are used for \( \infty \) because infinity is a concept, not a number, and cannot be reached or included.

Interval notation offers a neat and efficient way to express both finite and infinite sets in mathematics.

Solving Linear Inequalities

Solving linear inequalities is similar to solving equations, but with extra care around inequality signs. Here's what you should know:

• **Identify the Inequality:** A linear inequality looks like a regular equation but includes \(<, \leq, >,\) or \(\geq\).
• **Clear Fractions:** Multiply through by the least common denominator (LCD) to get rid of fractions. In our exercise, we used 6 since it's the LCD of 2 and 3.
• **Isolate the Variable:** Move all terms involving the variable to one side and constant terms to the other. In the example, subtracting \(2y\) and \(12\) simplified to \( y \geq -36 \).
• **Inequality Rule Alert:** If you multiply or divide both sides by a negative number, you must flip the inequality sign. This is a critical difference from equations.

Once simplified, the inequality is ready to be graphed and expressed in interval notation, confirming its solution set.