Problem 52 - Precalculus: Functions and Graphs

Question

Solve the equation or inequality. Express the solutions in terms of intervals whenever possible. $$|16-3 x| \geq 5$$

Step-by-Step Solution

Answer

The solution is \((-\infty, \frac{11}{3}] \cup [7, +\infty)\).

Step 1: Setup the Absolute Value Inequality

The given inequality is \(|16 - 3x| \geq 5\). This means the expression inside the absolute value, \(16 - 3x\), is either greater than or equal to 5, or less than or equal to -5.

Step 2: Break Into Two Separate Inequalities

The inequality can be separated into two cases:1. \(16 - 3x \geq 5\)2. \(16 - 3x \leq -5\)

Step 3: Solve the First Inequality

Start with \(16 - 3x \geq 5\):Subtract 16 from both sides: \(-3x \geq 5 - 16\)\(-3x \geq -11\)Divide each side by -3 and reverse the inequality sign:\(x \leq \frac{11}{3}\).

Step 4: Solve the Second Inequality

Now solve \(16 - 3x \leq -5\):Subtract 16 from both sides: \(-3x \leq -5 - 16\)\(-3x \leq -21\)Divide each side by -3 and reverse the inequality sign:\(x \geq 7\).

Step 5: Write the Solution in Interval Notation

The solution to the inequality \(|16 - 3x| \geq 5\) is the union of the two intervals found: \(x \leq \frac{11}{3}\) and \(x \geq 7\). In interval notation, this is written as:\((-\infty, \frac{11}{3}] \cup [7, +\infty)\).

Key Concepts

Inequality Solving

When dealing with inequalities, understanding the concept of absolute value is crucial. Absolute value inequalities often break down into two separate inequalities because the absolute value signifies how far a number is from zero. For example, with \( |16 - 3x| \geq 5\), we interpret this as the distance between \(16 - 3x\) and zero being at least 5. To solve this, we split it into two different inequalities: one where the expression inside is greater than or equal to 5 (\(16 - 3x \geq 5\)) and another where it is less than or equal to -5 (\(16 - 3x \leq -5\)). Remember, each inequality will be solved separately, but both contribute to the final set of solutions.Steps Involved:

• Identify that the absolute value inequality involves two possible scenarios based on the definition of absolute value.
• Translate this into two ordinary inequalities without absolute value signs.
• Solve each inequality for the variable.

Take careful note: when multiplying or dividing by a negative number, reverse the inequality sign to maintain the solution's correctness.

Interval Notation

Once you've solved a compound inequality and need to express solutions, interval notation is a concise method. It uses mathematical symbols to describe a set of numbers along a number line. In the solution \(x \leq \frac{11}{3}\) or \(x \geq 7\), we translate these into interval notation to easily demonstrate ranges of numbers.For \(x \leq \frac{11}{3}\), think of values stretching from negative infinity to the boundary point. This is expressed as \((-\infty, \frac{11}{3}]\), where the bracket ']' denotes inclusion of \(\frac{11}{3}\).Similarly for \(x \geq 7\), it includes all values greater than or equal to 7. Thus, we use \([7, +\infty)\), where '(' indicates excluding positive infinity.General Approach for Interval Notation:

• Use '(' or ')' if the endpoint is not included, and '[' or ']' if it is included.
• Always list intervals in increasing order from left to right.
• For infinite intervals, use \-\infty\ to denote extending left without bound and \+\infty\ to extend right without bound.

Combining different intervals requires the union operator '∪', signaling either interval's values are part of the solution.

Compound Inequalities

Compound inequalities involve the conjunction or disjunction of two distinct inequalities. When an inequality has an 'or' condition as in \(x \leq \frac{11}{3} \, \text{or} \, x \geq 7\), you essentially have scenarios covered by either inequality.Understanding Compound Inequalities:

• With 'or': Either of the conditions can contribute to the solution, and one does not restrict the other. This results in a union of solutions.
• For 'and', both conditions must be true simultaneously, limiting the solution to their overlap; however, this doesn't apply to our inequality, which uses 'or'.

In practical terms, solving \(|16 - 3x| \geq 5\) reveals that any x less than or equal to \(\frac{11}{3}\) or greater than or equal to 7 satisfies the inequality, leading to a union of two intervals. Use careful logic to interpret real-life situations where such inequalities apply, ensuring clarity and the correct application of both scenarios separately before combining their solutions.