Problem 53 - Beginning and Intermediate Algebra

Question

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$9 \leq|7-8 q|$$

Step-by-Step Solution

Answer

The solution to the inequality \(9 \leq|7-8 q|\) is \( q \in (-\infty, -\frac{1}{4}] \cup [2, \infty) \).

Step 1: Rewrite the absolute value inequality

Let's rewrite the inequality given: \( |7 - 8q| \geq 9 \) Since we're dealing with an absolute value, we can rewrite this equation as following two inequalities: \( 7 - 8q \geq 9 \) or \( 7 - 8q \leq -9 \)

Step 2: Solve each inequality individually

Now let's solve each inequality for q: 1. \( 7 - 8q \geq 9 \) Subtract 7 from both sides: \( -8q \geq 2 \) Now, divide by -8 (and remember to flip the inequality sign when dividing with a negative number): \( q \leq -\frac{1}{4} \) 2. \( 7 - 8q \leq -9 \) Subtract 7 from both sides: \( -8q \leq -16 \) Now, divide by -8 (and remember to flip the inequality sign again): \( q \geq 2 \)

Step 3: Write the solution in interval notation

We have found the two inequalities for q. Combine the solution in interval notation: \( q \in (-\infty, -\frac{1}{4}] \) for first inequality and \( q \in [2, \infty) \) for the second inequality As these intervals don't overlap, we can write the final solution: \( q \in (-\infty, -\frac{1}{4}] \cup [2, \infty) \)

Key Concepts

Linear Inequalities

Linear inequalities are expressions where variables undergo linear operations and are compared using inequality symbols like ">", "<", "≥", or "≤". The process of solving belongs to algebra, where we find the range of values that can satisfy the inequality conditions. For example, in the step-by-step solution, we dealt with the linear inequalities \( 7 - 8q \geq 9 \) and \( 7 - 8q \leq -9 \).
These linear inequalities often involve solving for a variable, such as \( q \), and require similar operations to solving linear equations: adding, subtracting, multiplying, or dividing by constants.
A critical rule to remember is when you multiply or divide within an inequality by a negative number, you must flip the inequality sign. This is exemplified in the calculations: dividing by \(-8\) changed \( \geq \) to \( \leq \) and vice versa.

Solution Set

A solution set is the collection of all values that satisfy a given inequality or equation. It represents the range where the variables meet the conditions set by the problem. For example, once we solve the inequalities \( 7 - 8q \geq 9 \) and \( 7 - 8q \leq -9 \), we obtain two ranges of values for \( q \): \( q \leq -\frac{1}{4} \) and \( q \geq 2 \).
This solution set expresses all possible \( q \) values meeting the conditions. Rather than a single point, solution sets can represent an entire interval or union of intervals, showing how diverse the answers to inequalities can be.

Interval Notation

Interval notation is a simple, shorthand method to describe the solution sets of inequalities. Instead of listing numbers, we use intervals to briefly capture all solutions.
For example, the inequalities \( q \leq -\frac{1}{4} \) is represented by the interval \((-\infty, -\frac{1}{4}]\), and the inequality \( q \geq 2 \) is represented by \([2, \infty)\).
The use of brackets indicates whether the endpoints are included or excluded:\

\
• "[ ]": the endpoint value is included in the solution (\( q = 2 \) is included in \([2, \infty)\).
\
• "( )": the endpoint value is not included in the solution (\( q eq -\frac{1}{4} \) for \((-\infty, -\frac{1}{4}]\).
\

Combining intervals like \((-\infty, -\frac{1}{4}] \cup [2, \infty)\) allows for the expression of more complex solution sets.

Absolute Value Equations

Absolute value equations are equations that involve the absolute value function \( |x| \), which measures the distance of a number from zero on a number line without considering direction. In simpler terms, it always gives a non-negative output.
When solving absolute value equations and inequalities like \( |7 - 8q| \geq 9 \), a typical approach is to transform them into two separate linear equations or inequalities. This is because an absolute value expression \( |x| = y \) is equivalent to \( x = y \) or \( x = -y \).
For inequalities, this translates to \( x \geq y \) or \( x \leq -y \), leading to two possible conditions that need to be evaluated separately, as seen in our solution transformation to \( 7 - 8q \geq 9 \) or \( 7 - 8q \leq -9 \). This allows us to capture all scenarios in which the absolute value condition is met.