Problem 53 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |-5 x+3|<12 $$

Step-by-Step Solution

Answer

The solution set is \((- \frac{9}{5}, 3)\) and the graph shows a shaded region between \(-\frac{9}{5}\) and \(3\) with open circles.

Step 1 - Understand the Absolute Value Inequality

The given inequality is \(|-5x + 3| < 12\). This means that the expression inside the absolute value must be less than 12 and greater than -12.

Step 2 - Break Down the Absolute Value Inequality

The inequality \(|-5x + 3| < 12\) can be split into two separate inequalities: \(-12 < -5x + 3 < 12\).

Step 3 - Solve the Inequality - Part 1

First, solve \(-12 < -5x + 3\). Subtract 3 from both sides: \(-12 - 3 < -5x\), which simplifies to \(-15 < -5x\). Divide by -5 and reverse the inequality sign: \(3 > x\), or \(x < 3\).

Step 4 - Solve the Inequality - Part 2

Now, solve \(-5x + 3 < 12\). Subtract 3 from both sides: \(-5x < 9\). Divide by -5 and reverse the inequality sign: \(x > -\frac{9}{5}\).

Step 5 - Combine the Solution

Combine the two inequalities: \(-\frac{9}{5} < x < 3\). This represents the solution set.

Step 6 - Graph the Solution Set

Draw a number line and shade the region between \(-\frac{9}{5}\) and \(3\), using open circles at both endpoints to indicate they are not included.

Step 7 - Write the Interval Notation

The interval notation for the solution set is \((- \frac{9}{5}, 3)\).

Key Concepts

Solving Inequalities

When solving inequalities, our main goal is to find all the values of the variable that make the inequality true. Unlike equations, inequalities do not restrict us to a single value; instead, they give us a range of values. Here's a simple step-by-step approach:

• Understand the inequality: Begin by understanding exactly what the inequality represents.
• Isolate the variable: Work towards getting the variable on one side of the inequality.
• Consider the direction: When you multiply or divide by a negative number, flip the inequality sign.

Applied to our example: The initial inequality \(|-5x + 3| < 12\) means we must consider the two possible outcomes: \(-12 < -5x + 3 < 12\). We can split this into two parts: \(-12 < -5x + 3\) and \-5x + 3 < 12\. Solving both inequalities step by step, always remember to flip the inequality sign when dividing by a negative.

Interval Notation

Interval notation is an efficient way to represent the solution set of an inequality. It uses parentheses and brackets to describe intervals of numbers.

• Parentheses () : Used for non-inclusive limits, meaning the endpoints are not included in the interval.
• Brackets [] : Used for inclusive limits, meaning the endpoints are included in the interval.

For our solution \(-\frac{9}{5} < x < 3\), we use parentheses because \-\frac{9}{5}\ and \3\ are not included in the solution. So the interval notation will be \((-\frac{9}{5}, 3)\). This notation succinctly communicates the range of x for which the original inequality holds true.

Graphing Solutions

Graphing the solution set of an inequality helps visualize the range of values that satisfy the inequality. Here's how to graph them on a number line:

• Draw a number line: Start with a horizontal line and mark the important points (endpoints of the interval).
• Open or closed circles: Use open circles to denote values that are not included, and closed circles for values that are included.
• Shade the range: Shade the portion of the number line that represents the solution set.

For our inequality \(-\frac{9}{5} < x < 3\), we draw a number line, place open circles at \-\frac{9}{5}\ and \3\, and shade the line between them, indicating that all values between these points satisfy the inequality.