Problem 57 - 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. $$|8-r| \leq 5$$

Step-by-Step Solution

Answer

The solution to the inequality \(|8-r| \leq 5\) in interval notation is: \([3, 13]\).

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

If the expression inside the absolute value is non-negative (i.e., (8 - r) ≥ 0), the inequality becomes: $$8 - r \leq 5$$ To solve for \(r\), subtract 8 from both sides:

Step 2: Simplify the inequality

$$-r \leq -3$$ Now, multiply both sides by -1. When we do that, we have to flip the inequality sign:

Step 3: Solve for \(r\) in Case 1

$$r \geq 3$$ In this case, r can be any value greater than or equal to 3. We can represent this in interval notation as: $$[3, \infty)$$

Step 4: Case 2: Expression inside the absolute value is negative

If the expression inside the absolute value is negative (i.e., (8 - r) < 0), we can multiply both sides by -1 (and remember to flip the inequality sign): $$ -(8 - r) \leq 5$$ Now, distribute the negative sign:

Step 5: Simplify the inequality

$$r - 8 \leq 5$$ Add 8 to both sides to solve for \(r\):

Step 6: Solve for \(r\) in Case 2

$$r \leq 13$$ In this case, r can be any value less than or equal to 13. We can represent this in interval notation as: $$(-\infty, 13]$$

Step 7: Combine the solutions

Now, since the original inequality was a "less than or equal to" inequality, we need to find the intersection of the solutions in both cases: $$[3, \infty) \cap (-\infty, 13]$$ To do this, we will take the greatest lower bound (3) and the least upper bound (13): $$[3, 13]$$ This interval represents all the values of \(r\) that will satisfy the original absolute value inequality.

Step 8: Answer

The solution to the inequality $$|8-r| \leq 5$$ in interval notation is: $$[3, 13]$$

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers on a continuum. It makes understanding the range of solutions simple and concise. For inequalities, we often use brackets:

• "[" or "]" implies that the number is included in the interval (closed interval).
• "(" or ")" means the number is not included (open interval).

For example, the solution to the inequality \( |8-r| \leq 5 \) is expressed as \([3, 13]\). This tells us that the range of possible values for \( r \) starts at 3 and ends at 13, inclusive of both endpoints.
Interval notation helps us quickly communicate the span of numbers in a very compact form.

Set Notation

Set notation is another way to describe the solution to equations and can be particularly useful for equations where you have distinct values rather than ranges. In set notation, a solution is enclosed in curly braces \( \{ \} \).

• It lists all elements that satisfy a certain condition.
• For example, if the solution to an equation were specific values like \( \{4, 10\} \), it would mean these are the exact values that work.

For inequalities involving absolute values, like the one given here, set notation might be less practical since it isn't a single fixed set of numbers but rather a continuous range.

Linear Inequalities

Linear inequalities describe a range of possible solutions for a variable. They are similar to linear equations but involve inequality signs such as \(\leq, \geq, <, >\). Solving them requires techniques similar to solving equations, with an additional rule: when multiplying or dividing by a negative number, flip the inequality sign.

• For example, from the step \(-r \leq -3\), multiplying by \(-1\) gives \(r \geq 3\), requiring flipping the inequality.
• The solutions can often result in multiple possible values of the variable that describe a range or interval.

Linear inequalities are foundational for understanding constraints in algebra.

Inequality Solution

An inequality solution describes a set of values that satisfy the inequality. It can often be visualized on a number line or within a context of a problem.

• Absolute value inequalities, like \(|8-r| \leq 5\), can split into two separate inequalities.
• Each inequality reflects the two cases formed by the absolute value condition: \((8-r)\) being positive and negative.
• After solving both inequalities separately, the results are combined, focusing on the overlapping solution sets.

In this scenario, the values of \( r \) satisfy both conditions \(r \geq 3\) and \(r \leq 13\), resulting in the final solution \([3, 13]\) in interval notation. This combination represents all potential values of \( r \) that meet the original absolute value inequality.