Problem 48 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$\frac{y}{4}-\frac{5}{8}<2$$

Step-by-Step Solution

Answer

The solution to the inequality is \(y < \frac{21}{2}\). The graph would contain an open circle at \(\frac{21}{2}\) with an arrow pointing to the left, indicating that y can be any value less than \(\frac{21}{2}\).

Step 1: Simplify the Inequality

First, we need to deal with the constant on the left side of the inequality, -5/8. To isolate y, we want to get rid of this constant. We do this by adding 5/8 to both sides of the inequality: \( \frac{y}{4} - \frac{5}{8} + \frac{5}{8} < 2 + \frac{5}{8} \), which simplifies to \( \frac{y}{4} < \frac{21}{8} \).

Step 2: Isolate y

Now, y is divided by 4, we can undo this by multiplying both sides of the inequality by 4: \( (\frac{y}{4}) \times 4 < (\frac{21}{8}) \times 4 \), yielding \( y < \frac{21}{2} \).

Step 3: Graph the Solution

Lastly, the inequality can be graphed. This is done on a number line. A circle is placed at the number that y is less than, \(\frac{21}{2}\), and an arrow goes to the left of this number, indicating that y can be any number less than \(\frac{21}{2}\). The circle is open since y can't be equal to \(\frac{21}{2}\).

Key Concepts

Solving Inequalities

Solving inequalities involves finding the set of values that make an inequality true, much like solving equations. However, inequalities indicate a range of possible solutions instead of one specific answer.
To solve an inequality such as \( \frac{y}{4} - \frac{5}{8} < 2 \), we follow similar steps as we do for solving equations:

• Isolate the variable: Just like in equations, the first step is to get the variable on one side of the inequality by itself. In our exercise, we started by eliminating \( \frac{5}{8} \) from the left side by adding it to both sides. This gives us \( \frac{y}{4} < \frac{21}{8} \).
• Undo operations: In the next step, eliminate any numbers or fractions attached to the variable. Here we multiply both sides by 4 to solve for \( y \), resulting in \( y < \frac{21}{2} \).

Be mindful when multiplying or dividing by negative numbers, as this causes the inequality sign to reverse. In our example, this isn't needed, but it's a critical rule to remember. By following these steps, you can solve any linear inequality with confidence.

Graphing Inequalities

Graphing inequalities on a number line helps you visualize the range of solutions. It's an essential part of understanding and demonstrating inequality solutions.
In our given problem, we need to graph \( y < \frac{21}{2} \). Here's how you can graph this inequality:

• Identify the critical value: Begin by marking the number \( \frac{21}{2} \) on the number line. This is the boundary point of the inequality.
• Use an open or closed circle: Since \( y \) is strictly less than \( \frac{21}{2} \), we use an open circle over this number to indicate it isn't included in the solution set.
• Draw the arrow: From the circle at \( \frac{21}{2} \), draw an arrow extending to the left, showing that \( y \) can be any number less than \( \frac{21}{2} \).

This simple graph visually represents the solution set for the inequality, making it easier to understand the range of possible values for \( y \). Remember, if the inequality were \( \leq \) or \( \geq \), a closed circle would indicate that the boundary value is included.

Algebraic Expressions

Algebraic expressions form the backbone of algebra, consisting of variables, numbers, and operational symbols like \(+\), \(-\), \(\times\), and \(\div\). In inequalities, expressions define one side of the inequality statement.
When working with the expression \( \frac{y}{4} \) within an inequality like \( \frac{y}{4} - \frac{5}{8} < 2 \), it's crucial to handle operations carefully.

• Understanding components: In our expression, \( \frac{y}{4} \) represents a fraction indicating that the number \( y \) is divided by 4.
• Simplify step-by-step: Just as we did, start by simplifying the expression to isolate the variable. This may involve combining like terms, reducing fractions, or using inverse operations.

These simplification skills are essential in manipulating algebraic expressions within equations and inequalities effectively. Mastery of this concept will aid in solving more complex algebraic problems seamlessly.