Problem 26 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ 3|2 x-2|+2 \geq-5 $$

Step-by-Step Solution

Answer

The solution is all real numbers, \(x \in \mathbb{R}\).

Step 1: Isolate the Absolute Value

The given inequality is \(3|2x-2| + 2 \geq -5\). First, subtract 2 from both sides to isolate the absolute value term:\[ 3|2x-2| \geq -7 \]

Step 2: Analyze Possibility of Solution

The expression \(3|2x-2| \geq -7\) is always true since the left side, being an absolute value multiplied by 3, is non-negative and \(-7\) is negative. This means the inequality is satisfied for all real numbers.

Step 3: Interpret the Solution

Since the inequality holds true for all real numbers, the solution set is all real numbers \(x\), represented as \(x \in \mathbb{R}\).

Step 4: Graph the Solution

On a number line, graph all real numbers by shading the entire line, which indicates that every real number is part of the solution set.

Key Concepts

Solution Set

When solving an inequality like \(3|2x-2| + 2 \geq -5\), the goal is to find the solution set, which consists of all values of \(x\) that make the inequality true. The process usually involves isolating the absolute value expression, analyzing it, and determining for which values it satisfies the inequality. In this example, we isolated the absolute value expression resulting in the inequality \(3|2x-2| \geq -7\). Observing that \(3|2x-2|\) is always non-negative and always greater than or equal to \(-7\) (since \(-7\) is negative and absolute values are always non-negative), we found that each and every real number satisfies this inequality. Hence, the solution set is all real numbers, often expressed as \(x \in \mathbb{R}\). It means that regardless of which real number \(x\) you choose, it will make the inequality true.

Graphing Inequalities

Graphing inequalities involving absolute values on a number line helps visualize the solution set. When the inequality is true for all real numbers, as in our problem, the entire number line represents the solution set.To graph this:

• Draw a horizontal line to represent the number line.
• Shade the entire line to indicate that all numbers on this line are part of the solution set.

By shading the entire number line, you effectively demonstrate that no matter what \(x\) value you choose along this line, it will satisfy the inequality \(3|2x-2| + 2 \geq -5\). This method of graphing provides a clear visual confirmation of the solution set.

Real Numbers

Real numbers form a continuous set of numbers that include both rational and irrational numbers. They can be positive, negative, or zero. In the context of inequalities, real numbers are the solutions we are typically interested in finding.In our inequality \(3|2x-2| + 2 \geq -5\), the solution set includes all real numbers, denoted as \(x \in \mathbb{R}\). This is because any real value of \(x\) will satisfy the inequality. Understanding which real numbers satisfy an inequality is crucial because real numbers can represent quantities and measurements in real-world applications, from distances to temperatures, and more. Recognizing when an inequality applies to all real numbers means acknowledging that the set encompasses infinite possibilities.