Problem 27 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. $$ \frac{3}{4} y \geq-2 $$

Step-by-Step Solution

Answer

The solution is \( y \geq -\frac{8}{3} \), represented by shading right of \(-\frac{8}{3}\) on a number line.

Step 1: Isolate the variable

The inequality we need to solve is \( \frac{3}{4} y \geq -2 \). To isolate \( y \), we first multiply both sides of the inequality by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \). This gives us: \[\frac{4}{3} \times \frac{3}{4} y \geq -2 \times \frac{4}{3}\]Simplifying the left side, we have \( y \geq -\frac{8}{3} \).

Step 2: Analyze the inequality direction

Since we multiplied or divided by a positive number, the direction of the inequality remains unchanged. Therefore, the solution to \( \frac{3}{4} y \geq -2 \) is \( y \geq -\frac{8}{3} \).

Step 3: Graph the solution set

To graph the inequality \( y \geq -\frac{8}{3} \), draw a number line. Plot a point at \(-\frac{8}{3}\) and shade the region to the right of this point to represent all numbers greater than or equal to \(-\frac{8}{3}\). Use a closed circle at \(-\frac{8}{3}\) since it is included in the solution set.

Key Concepts

Number Line Graphing

Number line graphing is a visual way of representing inequalities and their solutions. It helps you to easily see which numbers satisfy the inequality. For this inequality, we have \( y \geq -\frac{8}{3} \). To represent it:

• Draw a horizontal line and place numbers on it. This forms your number line.
• Locate \(-\frac{8}{3}\) on the number line. In most cases, it's between the integers \(-3\) and \(-2\).
• Mark this point with a closed circle. The circle is closed because \(-\frac{8}{3}\) is included in the solution set: \( y \geq -\frac{8}{3} \).

Shade the region to the right of \(-\frac{8}{3}\) to show all the values of \( y \) that satisfy the inequality. Every point you shade, including \(-\frac{8}{3}\) itself, represents a valid solution.

Inequality Solution Steps

The solution process for an inequality may seem similar to solving equations, but there are key differences. Here's how you solve an inequality like \( \frac{3}{4}y \geq -2 \):

• First, identify what you need to isolate - here it’s the variable \( y \).
• To do this, multiply by the reciprocal of the fraction in front of \( y \). So, multiply both sides by \( \frac{4}{3} \), the reciprocal of \( \frac{3}{4} \).
• This cancels out the fraction on the left side, transforming it to simply \( y \).
• Simplify the right side by multiplying: \(-2 \times \frac{4}{3} = -\frac{8}{3} \).

Now, you have the inequality: \( y \geq -\frac{8}{3} \), which is your solution. Notice the direction of the inequality didn’t change because we multiplied by a positive number.

Algebraic Manipulation

In solving inequalities, algebraic manipulation plays a crucial role. It involves rearranging and simplifying expressions to isolate the variable you're solving for. In this case with \( \frac{3}{4}y \geq -2 \):

• To manipulate the inequality effectively, you multiplied by the reciprocal of the coefficient of \( y \), which was \( \frac{3}{4} \).
• The reciprocal, \( \frac{4}{3} \), was used to get a clean solution: \( y \geq -\frac{8}{3} \).
• This step is critical because it helps to understand that multiplying or dividing by a positive number keeps the inequality sign the same.

Being careful with these manipulation steps ensures you find the correct solution without accidentally changing the inequality's meaning.

Graphing Solution Sets

Graphing solution sets for inequalities is about showing visually which numbers make the inequality true. For \( y \geq -\frac{8}{3} \):

• Your number line becomes a tool to highlight the range of possible solutions.
• Begin by marking the point \(-\frac{8}{3}\) and using a closed circle to include it in your graph.
• Shade to the right of \(-\frac{8}{3}\) because you’re highlighting all numbers \( y \) greater than or equal to \(-\frac{8}{3} \).
• This shading is important, as it communicates that every number in this region meets the condition \( y \geq -\frac{8}{3} \).

Using a number line for inequalities is straightforward. It makes complex algebraic solutions easier to digest and understand at a glance.