Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ \left|5 x-\frac{1}{2}\right|-\frac{3}{2} > 0 $$

Step-by-Step Solution

Answer

The solution set is \(x < -\frac{1}{5}\) or \(x > \frac{2}{5}\). Graph these intervals on a number line, shading the regions excluding the points \(-\frac{1}{5}\) and \(\frac{2}{5}\).

Step 1: Isolate the Absolute Value

To make the inequality easier to solve, we should begin by isolating the absolute value expression on one side. This is done by adding \(\frac{3}{2}\) to both sides of the inequality:\[\left|5x - \frac{1}{2}\right| > \frac{3}{2}.\]

Step 2: Set Up the Two Cases

The expression \(\left|5x - \frac{1}{2}\right| > \frac{3}{2}\) translates to two separate inequalities:1. \(5x - \frac{1}{2} > \frac{3}{2}\)2. \(5x - \frac{1}{2} < -\frac{3}{2}\).The absolute value inequality turns into these two linear inequalities.

Step 3: Solve the First Inequality

Solve the first inequality, \(5x - \frac{1}{2} > \frac{3}{2}\):Add \(\frac{1}{2}\) to both sides:\[5x > \frac{3}{2} + \frac{1}{2}\]Simplifying the right side gives:\[5x > 2\]Next, divide both sides by 5:\[x > \frac{2}{5}\].This inequality represents the first part of the solution set.

Step 4: Solve the Second Inequality

Now solve the second inequality, \(5x - \frac{1}{2} < -\frac{3}{2}\):Add \(\frac{1}{2}\) to both sides:\[5x < -\frac{3}{2} + \frac{1}{2}\]Simplifying the right side gives:\[5x < -1\]Next, divide both sides by 5:\[x < -\frac{1}{5}\].This inequality represents the second part of the solution set.

Step 5: Combine the Solutions

Combine the solutions from the previous steps:- \(x > \frac{2}{5}\)- \(x < -\frac{1}{5}\)The solution set, therefore, is the union of these intervals written as:\[x < -\frac{1}{5} \quad \text{or} \quad x > \frac{2}{5}.\]

Step 6: Graph the Solution on a Number Line

To graph the solution, draw a number line and shade the regions corresponding to \(x < -\frac{1}{5}\) and \(x > \frac{2}{5}\).- Place an open circle at \(-\frac{1}{5}\) as values closer to this, but to the left, are included.- Place another open circle at \(\frac{2}{5}\) as values closer to this, but to the right, are included.- Shade the portion of the number line to the left of \(-\frac{1}{5}\) and to the right of \(\frac{2}{5}\).

Key Concepts

Solution Set

In the world of inequalities, the solution set is crucial. It tells us all the possible values of the variable that make the inequality true. For the inequality \[\left|5 x-\frac{1}{2}\right| - \frac{3}{2} > 0\]first, we isolate the absolute value, resulting in two separate linear inequalities:- \[5x - \frac{1}{2} > \frac{3}{2}\]- \[5x - \frac{1}{2} < -\frac{3}{2}\]By solving them step-by-step, we end up with:- \(x > \frac{2}{5}\)- \(x < -\frac{1}{5}\)When we talk about the solution set here, we're referring to values of \(x\) that satisfy either of these conditions. So, the set is expressed as:- \(x < -\frac{1}{5}\) or \(x > \frac{2}{5}\)This result doesn't form a continuous block on the number line but rather consists of two separate portions.

Graphing Inequalities

Graphing inequalities is a visual way to understand the solution set. It allows us to see which values of a variable satisfy an inequality:- To graph the solution for \(x < -\frac{1}{5}\) or \(x > \frac{2}{5}\):

• Draw a number line.
• Mark \(-\frac{1}{5}\) with an open circle as values below this satisfy the inequality.
• Shade all the way to the left starting from \(-\frac{1}{5}\).
• Mark \(\frac{2}{5}\) with another open circle as values above this satisfy the inequality.
• Shade all the way to the right starting from \(\frac{2}{5}\).

The open circles indicate that the points themselves \(-\frac{1}{5}\) and \(\frac{2}{5}\) are not part of the solution set. The shaded areas on the number line show the range of values that make the original inequality true.

Linear Inequalities

Linear inequalities, like linear equations, involve linear expressions. The primary difference is the inequality sign replacing the equal sign. Solving linear inequalities is similar to solving equations but requires careful handling of inequalities:- When solving, perform operations as you would with equations:

• Add, subtract, multiply, or divide both sides of the inequality.
• When multiplying or dividing by a negative number, flip the inequality sign.

In our exercise, we derived:- \(5x - \frac{1}{2} > \frac{3}{2}\) leads to \(x > \frac{2}{5}\)- \(5x - \frac{1}{2} < -\frac{3}{2}\) leads to \(x < -\frac{1}{5}\)Handling these linear inequalities efficiently delivered us the separate solution set intervals of \(x\). They are an indispensable tool in various mathematical and real-world applications.