Problem 29 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ -0.6 y<-1.8 $$

Step-by-Step Solution

Answer

The solution is \(y > 3\) and the graph is an open circle at 3 with shading to the right.

Step 1: Isolate the variable

To solve the inequality \(-0.6y < -1.8\), you need to isolate \(y\). Start by dividing both sides of the inequality by \(-0.6\). Remember that when you divide or multiply both sides of an inequality by a negative number, you must flip the inequality sign. So, \(-0.6y/-0.6 > -1.8/-0.6\).

Step 2: Simplify the inequality

Perform the division on both sides. The left side simplifies to \(y\) as \(-0.6/-0.6 = 1\). For the right side, calculate \(-1.8/-0.6 = 3\). Thus, the inequality is now \(y > 3\).

Step 3: Graph the solution set on a number line

To graph \(y > 3\) on a number line, draw a line and mark a point at \(3\). Since \(y\) is greater than \(3\), draw an open circle at \(3\) to indicate that \(3\) is not included and shade to the right of \(3\) to represent all numbers greater than \(3\).

Key Concepts

Solving Inequalities

When solving inequalities, the main aim is to find the range of values that can satisfy the inequality condition. In the given inequality, \(-0.6y < -1.8\), we start by isolating the variable \(y\). This involves performing algebraic operations to simplify the inequality, just as we would with equations. However, there's a crucial difference: if we multiply or divide by a negative number, the inequality sign must be flipped. This means that in our example, dividing both sides by \(-0.6\) changes the inequality to \(y > 3\). It's important to remember that any incorrect handling of the inequality sign can lead to mistakes in identifying the valid solutions.

Graphical Representation

Graphing the solution set of an inequality provides a visual interpretation of where the solutions lie on the number line. For the inequality \(y > 3\), we want to display all numbers that are greater than 3. Begin by drawing a horizontal number line and marking a point at 3. Since the inequality does not include 3 itself, an open circle is used at this point to indicate non-inclusion. The region to the right of this open circle is shaded, showing that all numbers greater than 3 are solutions. This graphical representation is a helpful tool for understanding and communicating the solutions of inequalities more effectively.

Algebraic Manipulation

Algebraic manipulation involves using basic arithmetic operations and properties to simplify expressions or inequalities. In solving inequalities like \(-0.6y < -1.8\), we often follow a step-by-step process: isolate the variable, simplify expressions, and utilize properties of inequalities.

• Step 1: Isolate \(y\) by dividing both sides by \(-0.6\), which requires flipping the inequality sign due to division by a negative number.
• Step 2: Simplify the expression to obtain \(y > 3\).

Understanding these manipulations is vital for solving inequalities correctly. Properly applying these techniques is not just about reaching the solution but ensuring that each step logically follows from the last, maintaining the inequality's validity throughout the process.