Problem 59 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$|-3 x+8| \geq 3$$

Step-by-Step Solution

Answer

The solution is \(x \leq \frac{5}{3}\) or \(x \geq \frac{11}{3}\).

Step 1: Understand Absolute Value Inequalities

The given inequality is \(|-3x + 8| \geq 3\). Inequalities involving absolute value can be split into two separate inequalities: one for the positive value and one for the negative value. This is because the absolute value of any expression \(a\) is the distance of \(a\) from zero on the number line, thus \(|a| \geq b\) implies either \(a \geq b\) or \(a \leq -b\).

Step 2: Set Up Two Separate Inequalities

Based on the understanding from Step 1, we split the inequality \(|-3x + 8| \geq 3\) into two cases:1. \(-3x + 8 \geq 3\)2. \(-3x + 8 \leq -3\).

Step 3: Solve the First Inequality

For the first inequality, \(-3x + 8 \geq 3\):- Subtract 8 on both sides: \(-3x \geq 3 - 8\).- Simplify: \(-3x \geq -5\).- Divide both sides by -3, remembering to flip the inequality sign: \(x \leq \frac{5}{3}\).Thus, one solution to the inequality is \(x \leq \frac{5}{3}\).

Step 4: Solve the Second Inequality

For the second inequality, \(-3x + 8 \leq -3\):- Subtract 8 on both sides: \(-3x \leq -3 - 8\).- Simplify: \(-3x \leq -11\).- Divide both sides by -3, remembering to flip the inequality sign: \(x \geq \frac{11}{3}\).Thus, another solution is \(x \geq \frac{11}{3}\).

Step 5: Combine the Solutions

Both inequalities together imply that the solution set for the original inequality \(|-3x + 8| \geq 3\) includes values of \(x\) that satisfy either \(x \leq \frac{5}{3}\) or \(x \geq \frac{11}{3}\). This is a union of the two intervals.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship of one quantity being larger or smaller than another. Unlike equations, which state that two expressions are equal, inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to indicate the range within which a variable can lie.
When working with inequalities, especially those involving absolute values, it's important to understand that they describe a range, not just one solution.
In essence, an inequality provides us with a boundary or region on the number line. This range or region represents all possible solutions that satisfy the inequality condition.

• \(x > a\): All values of \(x\) greater than \(a\).
• \(x < a\): All values of \(x\) less than \(a\).
• \(x \geq a\): All values of \(x\) greater than or equal to \(a\).
• \(x \leq a\): All values of \(x\) less than or equal to \(a\).

Absolute value inequalities, such as \(|a| \geq b\), are intriguing because they split into two separate conditions. This splitting occurs due to the definition of absolute values which considers both positive and negative scenarios of a number.

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It provides a language through which we can model real-world scenarios using equations and inequalities.
One of the primary skills in algebra is solving equations and inequalities, which involves finding the value(s) for the unknown variables that satisfy the given condition.
In dealing with inequalities, algebraic manipulation is key. Steps like adding, subtracting, multiplying, or dividing both sides of an inequality by the same number are common

• Addition/Subtraction: You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality.
• Multiplication/Division: Multiplying or dividing both sides by a positive number keeps the inequality direction same, but if it's a negative number, the inequality direction flips.

For example, in the inequality \(-3x + 8 \geq 3\), you would subtract 8 from both sides and then divide by \(-3\), remembering to reverse the inequality sign due to division by a negative number.

Inequality Solving Steps

Solving an inequality involves a systematic approach to guarantee all possible solutions are identified. Here’s a simple guide to solving absolute value inequalities like \(|-3x + 8| \geq 3\):

• **Understanding the inequality:** Recognize that an absolute value inequality expresses variables under two scenarios—positive and negative.
• **Splitting the inequality:** For absolute value, create two separate inequalities without the absolute value. For \(|a| \geq b\), these are \(a \geq b\) and \(a \leq -b\).
• **Solving each inequality:** Perform algebraic operations to isolate the variable. Remember to reverse the inequality sign if dividing by a negative number. For instance:
  • \(-3x + 8 \geq 3\): Subtract 8, then divide by \(-3\) to get \(x \leq \frac{5}{3}\).
  • \(-3x + 8 \leq -3\): Subtract 8, then divide by \(-3\) to get \(x \geq \frac{11}{3}\).
• **Combining results:** Use these solved inequalities to find the set of all solutions, often resulting in a union of intervals. In this case, the solution is \(x \leq \frac{5}{3}\) or \(x \geq \frac{11}{3}\).

By following these structured steps, you ensure that you've accounted for all potential solutions, providing a complete and clear answer to the inequality problem.