Problem 37 - Beginning and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write the answer in interval notation. $$|8 k+5| \geq 0$$

Step-by-Step Solution

Answer

In this inequality, since an absolute value is always non-negative, the inequality \(|8k + 5| \geq 0\) is true for all k. The solution is \(k \geq -\frac{5}{8}\), and in interval notation, the solution set is \([- \frac{5}{8}, \infty)\].

Step 1: Case 1: The expression inside the absolute value is non-negative

We first consider the case where the expression inside the absolute value is non-negative, i.e., \(8k + 5 \geq 0\). In this case, the inequality becomes: $$|8k + 5| \geq 0 \Rightarrow 8k + 5 \geq 0$$ To solve for k, we just need to isolate k: $$8k \geq -5$$ Divide both sides by 8: $$k \geq -\frac{5}{8}$$

Step 2: Case 2: The expression inside the absolute value is negative

Now we consider the case where the expression inside the absolute value is negative, i.e., \(8k + 5 < 0\). In this case, the inequality becomes: $$|- (8k + 5)| \geq 0 \Rightarrow -(8k + 5) \geq 0$$ We solve for k: $$-8k - 5 \geq 0$$ $$-8k \geq 5$$ Divide both sides by -8 and remember to reverse the inequality sign: $$k \leq -\frac{5}{8}$$

Step 3: Combining the solutions

Notice that both cases lead us to the same inequality, \(k \geq -\frac{5}{8}\). Since an absolute value is always non-negative, the inequality \(|8k + 5| \geq 0\) is true for all k because the absolute value of a number is always greater than or equal to 0.

Step 4: Graph the solution set

To graph this solution, we draw a number line with k on the horizontal axis. Since k is greater than or equal to -5/8, we place a solid dot at -5/8 and then shade the number line to the right of -5/8 to indicate that k can be any number greater than or equal to -5/8.

Step 5: Write the answer in interval notation

Since the inequality includes all values of k greater than or equal to -5/8, the interval notation for the solution is: \[[-\frac{5}{8}, \infty)\]

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities involve finding the set of values that satisfy an inequality containing an absolute value expression. In this case, the expression is \(|8k + 5| \geq 0\). The absolute value \(|x|\) represents the distance of \(x\) from zero on the number line, so it's never negative. This means any expression, when within absolute value brackets, is always zero or positive.

• When solving inequalities like \(|8k + 5| \geq 0\), we realize it's always true, because absolute values are always zero or greater.
• The solution involves checking the conditions when the expression inside the absolute value is positive, zero, or negative to see their impact on the inequality.
• For such cases, the entire real number set often becomes a solution since the condition itself always holds true in all instances.

Graphing Absolute Value Inequalities

Graphing the solution of an inequality gives a visual way to understand all possible values that satisfy the inequality. Here, we look at \(8k + 5\) and determine the values of \(k\) that make \(|8k + 5| \geq 0\) true.

• Once you identify the inequalities for \(k\), depict these on a number line.
• In this example, since the inequality holds for all \(k\), the number line will be entirely shaded to indicate every number satisfies the condition.
• A closed dot at the starting point \(-\frac{5}{8}\) will show inclusivity in the solution, representing \(k\) can be equal to \(-\frac{5}{8}\) and any number greater.

This way, graphing inequalities helps in visually validating and explaining how solutions work and easily communicates the scope of solution sets.

Interval Notation Simplified

Interval notation is a concise way to represent sets of solutions or subsets of the number line. In solving \(|8k + 5| \geq 0\), interval notation efficiently captures the solution.

• Intervals are shown using brackets. A square bracket \([\) or \(]\) indicates the number is included in the set, while a parenthesis \((\) or \()\) indicates it's not included.
• For this inequality, all real numbers satisfy the inequality, starting from \(-\frac{5}{8}\), inclusive, to infinity.
• In interval notation, this is captured as \([-\frac{5}{8}, \infty)\), where \(-\frac{5}{8}\) is included, and infinity is never included, as it represents an idea, not a specific number.

Interval notation is a clean, mathematical shorthand to express solutions, especially helpful for understanding ranges at a glance.