Problem 62 - 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. $$|3 x+5|<17$$

Step-by-Step Solution

Answer

The solution of the inequality \(|3x+5|<17\) is (-22/3, 4].

Step 1: Rewrite the inequality

Rewrite the inequality without the absolute value, resulting in this pair of inequalities: \(3x+5<17\) and \(-(3x+5)<17\)

Step 2: Solve the system of inequalities

Solve each inequality separately. For \(3x+5<17\), subtract 5 from both sides to get \(3x<12\), then divide both sides by 3 to get \(x<4\). For \(-(3x+5)<17\), distribute the negative sign inside the parentheses to get \(-3x-5<17\). Add 5 to both sides to get \(-3x<22\), then divide both sides by -3 to get \(x>-22/3\). However, remember when you divide or multiply by a negative number, the sign of the inequality changes. So, the inequality becomes \(x>-22/3\)

Step 3: Graphing the Solution

Following the two solutions \(x<-22/3\) and \(x<4\), the values of \(x\) that satisfy both conditions lie in (-22/3,4]. Plot this on a number line.

Step 4: Interval Notation

Express the solution set in interval notation. This leads to (-22/3, 4].

Key Concepts

Understanding Inequalities Without Absolute Value

When dealing with inequalities without absolute value, it's important to grasp the basic principles of inequality solving. An inequality without an absolute value is a statement that relates two expressions with symbols like <, >, ≤, or ≥.

Solving these inequalities follows similar steps to solving regular equations: by adding, subtracting, multiplying, or dividing both sides of the inequality to isolate the variable. However, one critical point to remember is that when you multiply or divide by a negative number, you must flip the inequality sign.

A practical example would be solving the two inequalities resulted from removing the absolute value bars from our exercise:

• To solve for the inequality without absolute value, \(3x+5<17\), you would subtract 5 from both sides giving \(3x<12\) and then divide by 3 to isolate \(x\), finding the solution \(x<4\).
• The second inequality, after distributing the negative sign, becomes \-3x-5<17\, which simplifies to \(x>-\frac{22}{3}\) after adding 5 to both sides and dividing by -3 (changing the direction of the inequality sign).

By following these steps, we find the range of values for \(x\) that make the original inequality true.

Graphing Inequalities on a Number Line

When asked to graph inequalities, a number line is a valuable visualization tool. For the inequality \(x < 4\), you would draw a number line, locate the point corresponding to 4, and shade the region to the left since \(x\) is less than 4.

For the inequality \(x > -\frac{22}{3}\), you would find the point corresponding to -\frac{22}{3} on the number line and shade the region to the right since \(x\) is more than that value. The overlap between shaded regions of the two distinct inequalities represents the solution set to the original absolute value inequality.

If you have a closed dot or a bracket at the end of your shaded interval, it indicates that the endpoint is included in the solution (as in ≤ or ≥). An open dot or a parenthesis indicates that the endpoint is not included (as in < or >). In this exercise, \(x < 4\) ends with an open dot, whereas the other end of the interval, \(x > -\frac{22}{3}\), has a closed dot thanks to the inclusive greater than sign.

Expressing Solutions Using Interval Notation

The concept of interval notation provides a concise way to express the solution set of inequalities. It uses brackets and parentheses to display intervals of numbers that form the solution set.

In interval notation, a bracket [ or ] signifies that the endpoint is included in the set (≤ or ≥), while a parenthesis ( or ) signifies that the endpoint is not included (< or >).

The solution to our exercise in interval notation is \( (-\frac{22}{3}, 4] \). The parenthesis around -\frac{22}{3} indicates that this number is not part of the solution set since the inequality involving this number was strict (>). The bracket around 4 indicates that this endpoint is included because the inequality had a 'less than' sign with respect to 4, thus allowing for the possibility of \(x\) being exactly 4.

Interval notation is an efficient and widely accepted way to communicate the sets of numbers that satisfy a given inequality, and it's particularly useful in higher mathematics when dealing with complex ranges.