Problem 72 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|5 x-2|>13$$

Step-by-Step Solution

Answer

The solution to the inequality \(|5x-2|>13\) is \(x < -2.2\) or \(x > 3\), which in interval notation can be expressed as \((-∞, -2.2)U(3, ∞)\)

Step 1: Interpret the absolute value inequality

Interpret the absolute value inequality. \(|5x - 2| > 13\) can be rewritten as \(5x - 2 > 13\) and \(5x - 2 < -13\)

Step 2: Solve the inequalities

Solve the inequalities. For \(5x - 2 > 13\), add 2 to both sides to get \(5x > 15\). Then divide both sides by 5 to solve for \(x > 3\). For \(5x - 2 < -13\), add 2 to both sides to get \(5x < -11\). Divide both sides by 5 to get \(x < -\frac{11}{5}\) or \(x<-2.2\)

Step 3: Graph the solution on a number line

Use a number line to represent the solutions. The number line will have a hollow circle at x = -2.2 and at x = 3 (because the inequalities didn't include 'equal to'), an arrow pointing to the left from x=-2.2, and an arrow pointing to the right from x=3, indicating that x can be any value less than -2.2 or greater than 3)

Step 4: Express the solution in interval notation

Interval notation is another way to express the solution of an inequality. In this inequality, the solution can be written as \((-∞, -2.2)U(3, ∞)\), where 'U' stands for the union of the intervals, meaning 'or'

Key Concepts

Absolute Value

The concept of absolute value is fundamental in understanding inequalities. Absolute value, denoted as |a|, is the distance between the number 'a' and zero on the number line, regardless of direction. For instance, both |3| and |-3| equal 3 because both points are three units away from zero.

In the context of solving inequalities, the absolute value creates a scenario where there are two distinct cases to consider: When the expression inside the absolute value bars is positive, and when it's negative. Specifically, the inequality |5x - 2| > 13 states that the distance between '5x - 2' and zero must be greater than 13, which yields two separate inequalities: one where 5x - 2 is greater than 13 and another where it is less than -13.

Inequality Solutions

The step-by-step process of solving inequalities is designed to isolate the variable of interest. Starting with interpreting the absolute value inequality by considering the two possible cases, we translate an expression with absolute value into two separate inequalities without it.

• For the first case (5x - 2 > 13), we simply solve for 'x' as if it were a regular inequality.
• The second case (5x - 2 < -13) requires us to flip the inequality sign when we divide by a negative number.

These steps are crucial because they transform the compound inequality created by the absolute value into a format that can be solved with basic algebraic techniques. The solutions to these separate inequalities are where we will find the values for 'x' that satisfy the original absolute value inequality.

Interval Notation

Interval notation provides a succinct method to express sets of numbers, commonly used for the solutions of inequalities. In interval notation, we use parentheses or brackets to denote the boundaries of the interval and whether those boundaries are included as solutions (brackets) or not (parentheses).

In the given problem, since the inequality does not include the equality (indicated by '>' instead of '\geq'), we use parentheses. So the solution is expressed as \( (-\infty, -2.2) \cup (3, +\infty) \), where '-\infty' and '+\infty' signify that the intervals extend indefinitely in their respective directions and the union symbol '\cup' indicates that values from both intervals are part of the solution.

Number Line Graphing

Number line graphing is a visual representation of the solution to an inequality. It helps students understand where the possible values for 'x' are located relative to the number line.

Creating a Graph

When graphing our inequality solutions, hollow circles are used to indicate that the endpoints (in this case, x = -2.2 and x = 3) are not included in the solution set, while solid circles would be used if the endpoints were included. Arrows extend from these hollow circles to visually display the range of values for 'x' that satisfy the inequality. The arrows indicate that 'x' can be any number to the left of -2.2 or to the right of 3, but not including those numbers themselves.