Problem 76 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |2 x-5|-5>20 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -10) \cup (15, \infty)\).

Step 1: Isolate the Absolute Value

Begin by adding 5 to both sides of the inequality to eliminate the constant term outside the absolute value expression: \(|2x - 5| > 25\)

Step 2: Break Down the Absolute Value

The expression \(|2x - 5| > 25\) implies two separate inequalities because absolute value indicates distance:1. \(2x - 5 > 25\)2. \(2x - 5 < -25\)

Step 3: Solve the First Inequality

For \(2x - 5 > 25\):1. Add 5 to both sides: \(2x > 30\)2. Divide both sides by 2: \(x > 15\)

Step 4: Solve the Second Inequality

For \(2x - 5 < -25\):1. Add 5 to both sides: \(2x < -20\)2. Divide both sides by 2: \(x < -10\)

Step 5: Determine the Solution Set

Combine the solutions from both inequalities. Since \(x > 15\) and \(x < -10\), these regions are separate, forming two intervals. The solution set is:\((-\infty, -10) \cup (15, \infty)\)

Step 6: Graph the Solution Set

Draw a number line and shade the regions: 1. Open circle at -10 and shade leftwards to negative infinity. 2. Open circle at 15 and shade rightwards to positive infinity.

Key Concepts

Solution Set

When dealing with absolute value inequalities, it's crucial to understand the concept of the solution set. The solution set includes all possible values of the variable that satisfy the inequality. In our absolute value scenario, we began with the inequality \(|2x - 5| > 25\). This inequality can be broken down into two separate inequalities:

• \(2x - 5 > 25\)
• \(2x - 5 < -25\)

After solving each inequality separately, we found two distinct ranges for our variable, \(x\):

• \(x > 15\)
• \(x < -10\)

These ranges do not overlap, which means our solution set has two intervals. The outcome is a union of the two intervals: \((-2, -10) \cup (15, 2)\). The union symbol \(\cup\) indicates that the solution includes any value from either interval.

Interval Notation

Interval notation is a concise way of representing the solution set of an inequality. This notation uses brackets and parentheses to describe which parts of the number line are included in the solution set. In this case, our solution involves two open intervals:

• The interval \((-2, -10)\) represents all numbers less than -10. The parenthesis indicates that -10 is not included.
• The interval \((15, 2)\) represents all numbers greater than 15. Again, the parenthesis indicates 15 is not included in the set.

We use the union symbol \(\cup\) to connect these two intervals, indicating that both separate ranges are part of the solution. Interval notation offers a slick and efficient way to present complex solutions in a way that is easy to interpret.

Graphing Inequalities

Graphing inequalities on a number line can help you visualize the solution set. With the inequality \(|2x - 5| > 20\), we found two separate conditions:

• \(x > 15\)
• \(x < -10\)

To graph these, you'll draw a number line and use open circles to indicate that the endpoints (-10 and 15) are not part of the solution. For the inequality \(x < -10\), place an open circle at -10 and shade the line to the left, extending towards negative infinity. For \(x > 15\), put an open circle at 15 and shade the line to the right, extending towards positive infinity.
By shading these regions, you can clearly see that the solution includes all values less than -10 and all values greater than 15, confirming the intervals \((-2, -10)\) and \((15, 2)\). This visual representation complements the numerical approach and gives a clear picture of the solution set.