Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ |2-4 b| \leq 6 $$

Step-by-Step Solution

Answer

The solution set is \(-1 \leq b \leq 2\).

Step 1: Identify the Absolute Value Inequality

The absolute value inequality given is \(|2 - 4b| \leq 6\). The goal is to find the values of \(b\) (the solution set) that satisfy this inequality.

Step 2: Formulate Two Inequalities

For the absolute value \(|x| \leq a\), it translates to \(-a \leq x \leq a\). Apply this to the expression:\[-6 \leq 2 - 4b \leq 6\].

Step 3: Solve the Left Inequality

Start by solving the inequality \(-6 \leq 2 - 4b\):1. Subtract 2 from both sides to get \(-8 \leq -4b\).2. Divide each side by \(-4\) (remember to flip the inequality sign when dividing by a negative) to get \(2 \geq b\).

Step 4: Solve the Right Inequality

Now solve the inequality \(2 - 4b \leq 6\):1. Subtract 2 from both sides to obtain \(-4b \leq 4\).2. Divide each side by \(-4\) (flip the inequality sign again) to get \(b \geq -1\).

Step 5: Combine the Inequalities

The combined inequalities from the previous steps are:\[-1 \leq b \leq 2\].This represents the solution set for \(b\).

Step 6: Graph the Solution Set

Graph the inequality on a number line:- Draw a number line with points at \(-1\) and \(2\).- Use closed circles at \(-1\) and \(2\) because these points are included in the solution (they satisfy the \(\leq\) inequality).- Shade the region between \(-1\) and \(2\) to represent all values of \(b\) that satisfy the inequality.

Key Concepts

Understanding Inequality Solutions

Absolute value inequalities, like \(|2 - 4b| \leq 6\), are solved by breaking them into two separate inequalities. This involves understanding that the expression inside the absolute value can be both positive and negative, so we must consider both cases.

To solve \(|2 - 4b| \leq 6\), you set up two inequalities:

• -6 ≤ 2 - 4b
• 2 - 4b ≤ 6

By solving these inequalities step by step, you are able to determine the range of values for the variable \(b\) that satisfy the inequality. Essentially, you rearrange each inequality until you isolate \(b\) on one side. This solution set tells you exactly where on the number line \(b\) can land to still hold true to the original inequality. After solving, you combine your solutions to get \[-1 \leq b \leq 2\], were it shows everything \(b\) could be.

Graphing Inequalities

Once you have found the solution set, graphing these solutions helps visualize the range of values \(b\) can take. Graphing inequalities on a number line is straightforward. Here’s how you can accomplish that:

• Begin by drawing a horizontal line as your number line.
• Mark distinct points on the number line for each boundary, \(-1\) and \(2\), corresponding to the solution set \(-1 \leq b \leq 2\).
• Choose closed circles, also known as solid dots, since the inequality includes equal to (\( \leq \)).

Closed circles indicate that those precise points fall within the acceptable solution set, as the inequality implies that \(b\) can be equal to these boundary values.

Next, you need to represent all the potential values \(b\) could take, which is shaded in the interval between these marked points.

Number Line Representation of Inequalities

The number line representation of inequalities provides a visual cue that helps in understanding and interpreting solutions.

To represent the solution \(-1 \leq b \leq 2\) on a number line:

• Draw the number line and slightly exaggerate distances for clarity.
• Mark and label the key points \(-1\) and \(2\).
• Use closed circles on \(-1\) and \(2\) because these values are included in your solution set.
• Shade the segment of the number line that lies between \(-1\) and \(2\) to indicate all possible values for \(b\).

This shaded region signifies all the values that fulfill the inequality \(-1 \leq b \leq 2\).

If you imagine walking from point \(-1\) to point \(2\), all your possible steps cover every valid solution in the inequality's context. This visualization often makes inequalities more intuitive and easier to grasp.