Problem 81 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ |2-3 x| \geq-8 $$

Step-by-Step Solution

Answer

The solution is all real numbers: \((-\infty, \infty)\).

Step 1: Understanding Absolute Value Inequality

In this problem, we have the inequality \(|2 - 3x| \geq -8\). Note that since absolute values are always non-negative, \(|2 - 3x| \geq 0\) is always true for real numbers. Hence, \(|2 - 3x| \geq -8\) is automatically true because -8 is less than 0.

Step 2: Simplifying the Inequality to Always True

Since absolute values are always greater than or equal to zero, the inequality \(|2 - 3x| \geq -8\) holds for any real number. Therefore, there are no further steps needed for solving the inequality.

Step 3: Graph the Solution Set

The solution set for the inequality \(|2 - 3x| \geq -8\) is the set of all real numbers, since the inequality holds true for any real number \(x\). On a number line, this represents the entire line from negative infinity to positive infinity.

Step 4: Writing the Solution in Interval Notation

Since the inequality holds for all real numbers, we express the solution set in interval notation as \((-\infty, \infty)\).

Key Concepts

Graphing Inequalities

Graphing inequalities involves showing the range of values that satisfy a given condition. In the context of our problem with the inequality \(|2 - 3x| \geq -8\), we see that the inequality is valid for all real numbers. This happens because the absolute value expression is always zero or positive, hence it will certainly be greater than -8. To graph this on a number line, you would draw a line that extends infinitely in both directions from left to right.

When graphing, make sure to:

• Identify the range of values that satisfy the inequality.
• Shade or denote the appropriate section of the number line. Here, you would shade the entire line because all real numbers satisfy the inequality.
• Use open or closed circles to denote excluded or included boundary points. In this case, there aren't particular boundary points since the entire number line represents the solution.

This visual representation helps in understanding that \(|2 - 3x| \geq -8\) has no restrictions on the values of \(x\) within the real number system.

Interval Notation

Interval notation is a succinct way to express a range of numbers that satisfy an inequality. In our exercise, since the inequality \(|2 - 3x| \geq -8\) is true for all real numbers, we use the interval notation \((-\infty, \infty)\). This notation indicates every real number from negative infinity to positive infinity as part of the solution set.

Key points for using interval notation include:

• Parentheses \(()\) are used to show that endpoints are not included. Infinity symbols \((-\infty, \infty)\) always use parentheses because infinity is not a specific number we can include.
• Square brackets \([]\) would be used if the boundaries were included, such as \([a, b]\) showing \(a\) and \(b\) are part of the solution.
• Understanding that infinity and negative infinity are concepts rather than fixed points, so they can never be "included" in a set.

This notation streamlines communication of solution sets and helps convey information clearly to those familiar with the format.

Real Numbers

Real numbers comprise all the numbers on the number line, including both positive and negative integers, fractions, and irrational numbers like \(\sqrt{2}\) and \(\pi\). They are the building blocks for all sorts of calculations and equations. In the problem of \(|2 - 3x| \geq -8\), since \(-8\) is a negative number, and our inequality involves the absolute value, all possible values for \(x\) make the inequality true.

Real numbers are:

• Continuous: They form a seamless, unbroken span along the whole number line without any gaps.
• Inclusive: They include whole numbers and integers as well as decimals and fractions.
• Universal: In many scenarios, all calculations and algebraic manipulations happen within the universe of real numbers.

Understanding real numbers helps students to grasp why certain inequalities are valid for all values and aids them in visualizing number line solutions. As real numbers cover every conceivable quantity encountered in algebra, they are central to solving equations and graphing solutions.