Problem 64 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|-7 x-3| \geq 5$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{8}{7}] \cup [\frac{2}{7}, \infty)\).

Step 1: Understanding the Absolute Value Inequality

The inequality is given as \(|-7x-3| \geq 5\). An absolute value inequality of the form \(|A| \geq B\) can be broken down into two separate inequalities: \(A \geq B\) or \(A \leq -B\). This means we need to solve each of these individually.

Step 2: Solve First Inequality

For the first inequality, take \(-7x - 3 \geq 5\). To solve, first add 3 to both sides to obtain \(-7x \geq 8\). Then, divide each side by -7, remembering to flip the inequality sign: \(x \leq -\frac{8}{7}\).

Step 3: Solve Second Inequality

For the second inequality, take \(-7x - 3 \leq -5\). Add 3 to both sides to get \(-7x \leq -2\). Then, divide each side by -7, again flipping the inequality sign: \(x \geq \frac{2}{7}\).

Step 4: Combine Solutions

We solved two separate inequalities: \(x \leq -\frac{8}{7}\) and \(x \geq \frac{2}{7}\). Since this is an 'or' case (as given by the \(|A| \geq B\) format), the solution set includes any \(x\) that satisfies either condition. This means our solution is \((-\infty, -\frac{8}{7}] \cup [\frac{2}{7}, \infty)\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities are an interesting aspect of inequality problems. They involve expressions within absolute value bars, like \(|A| \), and often represent a distance from zero. In our problem, \(|-7x-3| \geq 5\), the expression inside the absolute value, \(-7x-3\), could be 5 steps or more away from zero in either direction on the number line.

To solve an absolute value inequality of the form \(|A| \geq B\), we must consider two scenarios:

• \(A \geq B\)
• \(A \leq -B\)

This dual approach is necessary because absolute values define distance in both positive and negative directions. By solving these two sets of inequalities, we can find all possible values of \(x\) that satisfy the original inequality.

Interval Notation

Interval notation is a succinct way of expressing a range of values, and it's particularly useful when working with inequalities. When expressing solutions in interval notation, we need to remember a few key rules:

• Parentheses, \(( \; )\), indicate that the endpoint is not included.
• Brackets, \([ \; ]\), mean the endpoint is included.
• The infinity symbols, \(+\infty\) and \(-\infty\), are always used with parentheses, \(( \; )\), because they are not actual numbers.

In our example, after solving \(|-7x-3| \geq 5\), we found the solution as two separate intervals:

• \(( -\infty, -\frac{8}{7}]\), signifying all numbers less than or equal to \(-\frac{8}{7}\)
• \([\frac{2}{7}, \infty)\), indicating all numbers greater than or equal to \(\frac{2}{7}\)

These intervals are then combined using the union symbol, \(\cup\), to express that the solution includes values from both intervals.

Solving Inequalities

Solving inequalities is all about finding the set of values that make the inequality true. Inequalities, such as \(A > B\) or \(A \leq B\), are solved similarly to equations but with a crucial point: whenever we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign.

Let's break down the solution for the given inequality \(|-7x-3| \geq 5\):

• Step 1: Remove the absolute value by setting up two cases:
  • \(-7x-3 \geq 5\) leading to \(x \leq -\frac{8}{7}\)
  • \(-7x-3 \leq -5\) resulting in \(x \geq \frac{2}{7}\)
• Step 2: Solve each inequality separately, paying careful attention to the direction of the inequality when dividing by a negative number.

Finally, consider that this is an 'or' situation due to the structure of the original absolute value inequality, so the solution includes any \(x\) satisfying either inequality. Understanding these steps ensures you can tackle any similar inequality problems with confidence.