Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(\left|1-\frac{2 x}{3}\right|<1\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 0 \), and the graph has an open circle at 0, shaded to the right to include all values greater than 0.

Step 1: Understand the properties of absolute value

The absolute value of a number is its distance from zero. Therefore, the absolute value of any number can be either positive or zero, but never negative. We can exploit this property to split the inequality into two separate inequalities.

Step 2: Split the inequality

According to the properties of absolute value, we can rewrite \( \left|1-\frac{2 x}{3}\right| < 1 \) as two separate inequalities: \(1-\frac{2x}{3} < 1\) and \( \frac{2x}{3} -1 > -1\).

Step 3: Solve the inequalities

Solving these inequalities, from the first inequality, when we subtract 1 from both sides, we get \(-\frac{2x}{3} < 0 \) which simplifies to \(x > 0\). Similarly, from the second inequality, adding 1 to both sides, we get \(\frac{2x}{3} > 0\), which simplifies to \( x > 0 \).

Step 4: Graph the solution

On a real number line, plot an open circle at 0, and shade to the right of 0, to represent all values greater than 0.

Key Concepts

Solving Absolute Value Inequalities

Understanding how to solve absolute value inequalities is crucial for mastering the concepts of algebra. The absolute value of a number represents its distance from zero on a number line, regardless of direction. When you're presented with an absolute value inequality, like \(\left|1-\frac{2 x}{3}\right|<1\text{,}\) it tells you that the expression inside the absolute value is less than 1 unit away from zero. To find the solution, we split this into two separate scenarios: one where the expression is positive (\(1 - \frac{2x}{3} < 1\)) and one where it is negative (\(1 - \frac{2x}{3} > -1\text{.})\) These inequalities are then solved individually to find the values of x that satisfy both conditions.
When approaching such problems, always keep in mind to consider both the positive and the negative versions of the number inside the absolute value. This dual consideration will guide you to the correct range (or ranges) of solutions.

Graphing Inequalities on a Number Line

To visualize the solutions of inequalities, graphing on a number line is extremely helpful. With absolute value inequalities, we often end up with a range of solutions which is best represented on this line. Consider the inequality \(\left|1-\frac{2 x}{3}\right|<1\text{.)}\) After solving it and obtaining \(x > 0\text{,}\) we graph this on a number line by plotting an open circle at 0 to indicate that 0 is not included in the solution and shading the line to the right to represent all values that are greater than 0.

Tip for Graphing:

• An open circle is used for inequalities with '<' or '>' which exclude the boundary point.
• A closed circle is used for inequalities with '\leq' or '\geq' which include the boundary point.
• Shading the number line to the right signifies values greater than the point, while shading to the left signifies values less than the point.

Graphing on a number line not only aids in understanding the range of solutions but also provides a visual verification of the inequality's truthfulness.

Properties of Absolute Value

The properties of absolute value are fundamental to solving equations and inequalities involving absolute values. At its core, the absolute value represents the magnitude of a real number without regard to its sign. Here are some critical aspects to remember:

• The absolute value of a number is always non-negative.
• If the absolute value is set less than zero, the inequality has no solution as a number's magnitude cannot be negative.
• The equation \(\left| x \right| = a\) with \(a \geq 0\) has two solutions: \(x = a\) and \(x = -a\text{.)}\) The same modulus value may come from a positive or negative origin.
• When an absolute value is set greater than a positive number, it typically results in a compound inequality reflecting a range of values that do not include the value within the absolute value.

Recognizing these properties allows for a systematic approach to solving absolute value inequalities. The sign of the number inside the absolute value defines the direction of the inequality once split into a compound inequality.

Inequality Solution Sets

The solution set of an inequality is the collection of all values that satisfy the inequality. When dealing with absolute value inequalities, the solution set can sometimes consist of two separate intervals, as these inequalities can have more than one range of solutions. It's essential to determine these intervals by solving the inequality twice, once for the positive case and once for the negative case.

Identifying Solutions Sets:

• Always consider the direction of the inequality. This will determine whether you are looking for values above or below certain points on the number line.
• For a single-variable linear inequality, the solution set is often an interval or a union of intervals displayed using interval notation.

In our original exercise \(\left|1-\frac{2 x}{3}\right|<1\text{,}\) once we determine the set of valid x-values, we combine them to present a complete solution set. In this case, we’ve found that \(x > 0\) is our solution set, which is an interval starting just above 0 and extending indefinitely to the right on the number line.