Problem 100 - College Algebra Essentials

Question

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=|2 x-5|+1\) and \(y>9\).

Step-by-Step Solution

Answer

The solution for \(x\) in interval notation is \((-\infty, -1.5) \cup (6.5, \infty)\).

Step 1: Isolate the Absolute Value

To solve for \(x\), you first need to isolate the absolute value term. Doing so, subtract 1 from both sides of the inequality: \(|2x-5| > 9 - 1\), which simplifies to \(|2x - 5| > 8\).

Step 2: Remove the Absolute Value

Now that the absolute value term is isolated, you can remove the absolute value sign. The definition of absolute value states that \(|a| = a\) or \(|a| = -a\). Therefore, considering both the positive and negative possibilities, the original inequality \(|2x - 5| > 8\) can be written as two separate inequalities: \(2x - 5 > 8\) and \(2x - 5 < -8\).

Step 3: Solve for x

Solve for \(x\) for each of these inequalities. In the first inequality \(2x - 5 > 8\), add 5 to each side to isolate \(x\), yielding \(2x > 13\), then divide each side by 2, resulting in \(x > 6.5\). Similarly, in the other inequality \(2x - 5 < -8\), add 5 to both sides to get \(2x < -3\), then divide each side by 2 to obtain \(x < -1.5\).

Step 4: Write the Solution in Interval Notation

Interval notation is a way of writing subsets of the real number line. A greater than or less than inequality is represented with parentheses. Therefore, the solution to the inequalities, \(x > 6.5\) and \(x < -1.5\), in interval notation is \((-\infty, -1.5) \cup (6.5, \infty)\). This means that \(x\) is either less than -1.5 or greater than 6.5.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that use the absolute value operator, which denotes the distance of a number from zero on a number line, irrespective of its direction. This can be thought of as the non-negative part of any real number, expressed as

• \(|a| = a\) if \(a\) is positive or zero.
• \(|a| = -a\) if \(a\) is negative.

In the context of inequalities, the task is often to express two scenarios: one where the expression inside is greater and the other, where it is lesser by the inequality's negation. For example,

• If \(|2x-5| > 8\), this results in two separate inequalities: \(2x-5 > 8\) and \(2x-5 < -8\).

This reflects the absolute value concept where you address both the positive and negative spans of the term within the absolute value.

Solving Inequalities

Inequalities are mathematical statements that denote one side of an equation is not necessarily equal to the other but rather larger, smaller, or equal under not exact conditions. Solving these involves finding the value ranges that satisfy the inequality. The basic steps include:

• Isolating the variable of interest, keeping inequality symbol direction intact.
• Performing arithmetic operations similar to equations but ensuring rules specific to inequalities are adhered to, like flipping the inequality sign when multiplying or dividing by a negative number.
• For an absolute value inequality like \(|2x-5| > 8\), solve \(2x-5 > 8\) and \(2x-5 < -8\), resulting in two inequalities to solve.
• Apply these to isolate \(x\):\(2x > 13\) gives \(x > 6.5\), while\(2x < -3\) results in \(x < -1.5\).

This way, you find all possible solutions for the inequality across the real number set.

Real Number Line

The real number line is a continuum of numbers that includes both rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). Representing solutions on a number line helps visualize the range or interval of values that satisfy an inequality or an equation.
For instance, if you have an inequality solution such as \(x > 6.5\), once plotted on the number line, it spans everything beyond 6.5 towards infinity, not including 6.5 itself. The number line is crucial for understanding interval notation as it visually denotes where solutions lie relative to one another in a spatial sense on a continuum.
In the context of our earlier examples, representing \(x < -1.5\) and \(x > 6.5\) clearly on the line shows two separate regions that fulfill our inequality conditions from the interval notation \((-\infty, -1.5) \cup (6.5, \infty)\).

Compound Inequalities

Compound inequalities involve combining two or more inequalities into a single expression that captures multiple conditions. They are often joined by "and" or "or."

• "And" indicates all conditions must simultaneously be true, typically written as a range — for instance, \( -1 < x < 5 \).
• "Or" signifies that satisfying any one of several conditions is sufficient. In our case from the absolute value inequality \(|2x-5| > 8\), we derived \(x < -1.5\) or \(x > 6.5\).

Intervals can be depicted as separate or overlapping. The interval notation \((-\infty, -1.5) \cup (6.5, \infty)\) exemplifies this concept, showcasing two separate ranges on the real line, indicating separate solutions satisfying the compound condition. Compound inequalities allow for more complex and complete representations of possible solutions.