Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(|9-2 x|-2<-1\)

Step-by-Step Solution

Answer

The solution set is \( 4 < x < 5 \).

Step 1: Rearrange the inequality

Firstly, add 2 to both sides of the inequality to isolate the absolute value part. The inequality becomes \( |9-2x| < -1 + 2 \) or \(|9-2x| < 1 \).

Step 2: Split the inequality

To solve this absolute value inequality, split it into two separate inequalities. The resultant inequalities are \( 9-2x<1 \) and \( 9-2x>-1 \)

Step 3: Solve for x in both inequalities

Solving the first inequality yields: \( 9-2x<1 \) -> \( -2x <1-9 \) -> \( -2x < -8 \) -> \( x > 4 \). Now, solving the second inequality: \( 9-2x>-1 \) -> \( -2x > -1-9 \) -> \( -2x > -10 \) -> \( x < 5 \).

Step 4: Determine the solution set and graph it

The solution set for this inequality is the set of all x such that \( 4 < x < 5 \). To graph this on the real number line, locate points for 4 and 5. Since 4 and 5 are not included in the solution, open circles are used at these points. Then, shade the region between these two points which represents all numbers between 4 and 5.

Key Concepts

Absolute Value Graphing

Understanding how to graph absolute value inequalities is a crucial skill in mathematics. The absolute value of a number is its distance from zero on the number line, regardless of direction. So, it's always a non-negative number. When graphing, you'll typically be dealing with two scenarios – where the absolute value is less than a number (creating an 'inside' solution set) or greater than a number (an 'outside' solution set).

For the exercise at hand, after isolating the absolute value expression, we have to consider both less than and greater than scenarios. Graphing these on a coordinate plane generally involves V-shaped graphs opening upwards or downwards. However, with inequalities, we're focused on a one-dimensional number line. You'll mark the critical points – the values that make the inside of the absolute value zero – and shade either inside or outside these points depending on the inequality. In this case, the inequality solution, consisting of all numbers between 4 and 5, is indicated by an open circle at these points and shading in-between, reflecting that these endpoints are not included.

Inequality Solution

Solving inequalities differs from equations as the solution is often a range or set of numbers rather than a single value. When the inequality involves an absolute value, it usually results in two separate inequalities, without the absolute value, to consider. It's a pivotal step, as it addresses the 'two-sided' nature of absolute values – reflecting the distance on both sides of zero on the number line.

In our example, we first balanced the inequality by isolating the absolute value on one side, leading to a comparison with 1. From there, we created and solved two linear inequalities, resulting in the solution set where x is between 4 and 5. Remember, understanding the rules of inequality manipulation is vital. Reversing the inequality sign when multiplying or dividing by a negative number is one such important rule to keep in mind when finding your solution.

Real Number Line Representation

Representing solutions to inequalities on a real number line allows us to see the set of all possible solutions at a glance. A number line is a visual tool that shows numbers as points on a line, with the position based on their value. For the absolute value inequality we are examining, the solution is shown as a segment on the number line instead of discrete points.

In practice, an open circle is used to denote that a number is not included in the set (denoting 'less than' or 'greater than'), while a closed circle signifies that the number is included ('less than or equal to' or 'greater than or equal to'). A common mistake is to forget to flip the inequality sign after multiplying or dividing by a negative number during the solution process. In our exercise, the number line will display open circles at 4 and 5 with a shaded region between them, symbolizing all real numbers greater than 4 and less than 5—a handy representation of the range of solutions to the inequality.