Problem 44 - College Algebra with Modeling and Visualization

Question

Solve each equation or inequality. (a) \(|5-x|=2\) (b) \(|5-x| \leq 2\) (c) \(|5-x| \geq 2\)

Step-by-Step Solution

Answer

(a) \(x = 3\) or \(x = 7\). (b) \([3, 7]\). (c) \((-\infty, 3] \cup [7, \infty)\).

Step 1: Recognize the Absolute Value Equation

We need to solve the equation \(|5 - x| = 2\). This absolute value equation will have two possible scenarios which arise from the definition of absolute value.

Step 2: Solve First Scenario for Equality

The absolute value of an expression equals a number when the expression inside the absolute value is either equal to the positive value of that number or the negative. For \(|5 - x| = 2\), we set up the first scenario: \(5 - x = 2\). Solving for \(x\), we subtract 5 from both sides to get \(-x = -3\), then multiply by -1 to find \(x = 3\).

Step 3: Solve Second Scenario for Equality

The second scenario is when the expression inside the absolute value is equal to the negative of that number: \(5 - x = -2\). Solving for \(x\), we subtract 5 from both sides to get \(-x = -7\), then multiply by -1 to find \(x = 7\).

Step 4: Expression of Solutions for Equality

Both solutions from the scenarios in the absolute value equation \(|5 - x| = 2\) hold true, meaning \(x = 3\) or \(x = 7\) is the solution set for the equation.

Step 5: Recognize the Absolute Value Inequality

We now approach the inequality \(|5 - x| \leq 2\). Absolute value inequalities require dividing the inequality into a compound inequality.

Step 6: Translate to Compound Inequality for \(\leq\)

For \(|5 - x| \leq 2\), transform into a compound inequality: \(-2 \leq 5 - x \leq 2\).

Step 7: Solve Left Part of Compound Inequality

We start solving \(-2 \leq 5 - x\). Subtract 5 from both sides: \(-2 - 5 \leq -x\) leads to \(-7 \leq -x\). Multiply by -1 (remember to flip the inequality sign) to get \(x \leq 7\).

Step 8: Solve Right Part of Compound Inequality

Now solve \(5 - x \leq 2\). Subtract 5 from both sides: \(-x \leq -3\). Multiply by -1 (flip inequality) to get \(x \geq 3\).

Step 9: Combine Solutions for \(\leq\) Inequality

Putting our compound inequality together from steps 7 and 8, we have: \(3 \leq x \leq 7\), or \(x\) is in the interval \([3, 7]\).

Step 10: Recognize the Absolute Value Inequality for \(\geq\)

Next, consider the inequality \(|5 - x| \geq 2\). By the nature of absolute values, this inequality splits into two part.

Step 11: Set Up Compound Inequalities for \(\geq\)

This absolute inequality has two cases: 1. \(5 - x \geq 2\)2. \(5 - x \leq -2\)

Step 12: Solve First Part for \(\geq\) Inequality

Solving \(5 - x \geq 2\) gives a similar process: Subtract 5 from both sides to obtain:\(-x \geq -3\), multiply by -1 (remember flipping the inequality), \(x \leq 3\).

Step 13: Solve Second Part for \(\geq\) Inequality

For \(5 - x \leq -2\), subtract 5: \(-x \leq -7\), multiply by -1 (flip inequality),\(x \geq 7\).

Step 14: Combine Solutions for \(\geq\) Inequality

The solution is the union of these intervals, so \(x \leq 3\) or \(x \geq 7\). This can be expressed as \(x \in (-\infty, 3] \cup [7, \infty)\).

Key Concepts

Inequalities

Inequalities are expressions that relate two values by showing that one is greater than, less than, or not equal to another. When tackling inequalities, it's important to understand their symbols like \(<\), \(>\), \(\leq\), and \(\geq\). These symbols help us determine the range of values that satisfy an inequality.
In problems involving absolute value inequalities, you often deal with two conditions. This is because the absolute value expression can affect the inequality in different ways depending on its positive or negative nature. Translating these into mathematical statements allows us to solve them by considering multiple scenarios.
For example, in solving \(|5-x| \leq 2\), we explore two conditions: \(5 - x \leq 2\) and \(5 - x \geq -2\). By solving each inequality separately, we can find the complete set of solutions that satisfy the given condition.

Compound Inequality

A compound inequality combines two inequalities within one statement. These inequalities can either use \("and"\) or \("or"\) to join conditions that represent the solution set.
In the example \(|5-x| \leq 2\), we are dealing with a compound inequality expressed as \(-2 \leq 5-x \leq 2\). This uses the \("and"\) condition, which means the expressed solution must satisfy both parts: greater than or equal to -2 and less than or equal to 2 simultaneously.
This method of ‘splitting’ inequalities helps to determine a range within which all values of \(x\) will satisfy all inequalities at once. Thus, solving these compound inequalities effectively helps students understand and find solutions for more complex problems involving range and boundaries.

Interval Notation

Interval notation is a powerful tool used to succinctly express a range of values that satisfy an inequality. It offers a visual and practical way to capture parts of the number line where conditions hold true.
For example, if we have the solution \(3 \leq x \leq 7\), we can write this solution in interval notation as \([3, 7]\). The square brackets signify that the endpoints are included, meaning \(x\) can be exactly 3 or 7.
Similarly, for the compound inequality \(-\infty < x \leq 3\) or \(x \geq 7\), the interval notation becomes \((-fty, 3] \cup [7, \infty)\). Here, a parenthesis \(()\) implies that the endpoint is not included, which often happens with \(\infty\) as it represents an unbounded limit. This notation not only simplifies our solutions but also enhances understanding of which values are acceptable.