Problem 64 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|3(x-1)+2| \leq 20$$

Step-by-Step Solution

Answer

The solution set is \([-4,8]\).

Step 1: Rewrite without absolute value

Recall that the definition of \(|f(x)| \leq k\) is \(-k \leq f(x) \leq k\). Use this to write the absolute value inequality as a compound inequality: \(- 20 \leq 3(x-1)+2 \leq 20\).

Step 2: Simplify the inequalities

Simplify both inequalities individually. The simplified form of the left inequality is \(x \geq -4\), and the right inequality is \(x \leq 8\). Solve each inequality on their own to find two potential roots, these will be the end points of the interval solution.

Step 3: Graph the solution on a number line

To graph the solution set, draw a number line and mark the two end points obtained from step 2, which are -4 and 8. Shade the interval between the two points inclusively.

Step 4: Express the solution using interval notation

The interval notation for graphed solution, which includes all x such as \(-4 \leq x \leq 8\), is \([-4,8]\). This denotes all real numbers that lie in between -4 and 8, including both -4 and 8.

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities joined by the terms "and" or "or." In this case, we are dealing with an "and" type situation because we want both conditions of the inequality to be true simultaneously. This type of inequality helps in situations where we need to find values that satisfy more than one condition. In the given exercise, once we remove the absolute value bars, we transform \( |3(x-1)+2| \leq 20 \) into the compound inequality \(-20 \leq 3(x-1)+2 \leq 20\).

• The term \("-20 \leq 3(x-1)+2"\) represents one inequality.
• "And" \( "3(x-1)+2 \leq 20" \) represents the second inequality. Both need to be true at the same time.

After solving these, we find that \( x \geq -4 \) and \( x \leq 8 \). This implies that the set of numbers that satisfy the original inequality form a continuous range between these two values.
Compound inequalities like these are often represented by shading in the region between two points on a number line, and they are fundamental when working with absolute value inequalities.

Interval Notation

Interval notation is a concise way to describe a set of numbers within a certain range. It uses parentheses \(( )\) and square brackets \([ ]\) to indicate whether endpoints are included or excluded:

• Square brackets, \([\text{ or }]\), are used to include an endpoint.
• Parentheses, \((\text{ or })\), are used if an endpoint is excluded.

For the solution of the compound inequality from our exercise, \(-4 \leq x \leq 8\), the interval notation is \([-4, 8]\). This tells us that all numbers between -4 and 8, including -4 and 8 themselves, are solutions.
Interval notation is especially useful because it provides a clear and efficient way to represent solution sets, especially when dealing with a large number of solutions. It allows us to easily communicate which values are included in a solution, eliminating ambiguity.

Graphing Inequalities

Graphing inequalities is a visual representation of the solution set of an inequality on a number line. This is particularly helpful in understanding the range of solutions available and ensuring that they satisfy the inequality's conditions. The process involves:

• Identifying the key numbers, or endpoints, from the simplified inequalities.
• Marking these numbers on a number line.
• Shading the region between these numbers when both are included.
• Using open circles on the endpoints for "less than" or "greater than" inequalities, and closed circles for "less than or equal to" or "greater than or equal to."

In the exercise, after determining that \(-4 \leq x \leq 8\), we graph it with a number line. Place closed circles on -4 and 8, because the inequality includes these points ("equal to"). The space between -4 and 8 is shaded to show that all those points form part of the solution set.
Graphing provides an intuitive understanding of inequalities and clearly illustrates what numbers satisfy the inequality. This visual aid can greatly enhance comprehension, particularly when learning about absolute value and compound inequalities.