Problem 52 - College Algebra with Modeling and Visualization

Question

Solve the equation symbolically. Then solve the related inequality. $$|x|-10=25, \quad|x|-10<25$$

Step-by-Step Solution

Answer

The equation solutions are \(x = 35\) and \(x = -35\). The inequality solutions are \(-35 < x < 35\).

Step 1: Solve the Equation

Start by solving the equation \(|x| - 10 = 25\). Add 10 to both sides:\[|x| = 35\]This means that \(x\) can either be \(35\) or \(-35\). So, the solutions to the equation are:\[x = 35 \quad \text{or} \quad x = -35\]

Step 2: Set Up the Inequality

Now solve the inequality \(|x| - 10 < 25\). Start by adding 10 to both sides of the inequality:\[|x| < 35\]

Step 3: Solve the Inequality

The inequality \(|x| < 35\) means that the values of \(x\) must satisfy:\[-35 < x < 35\]This gives us an interval of solutions for the inequality.

Key Concepts

Inequality Solving

When solving inequalities, the goal is to find all possible values of a variable that make the inequality true. For the inequality \(|x| - 10 < 25\), we start by isolating the absolute value term. This is done by adding 10 to both sides of the inequality, resulting in \(|x| < 35\). The expression \(|x| < 35\) suggests that the distance of \(x\) from zero must be less than 35. In simpler terms, \(x\) can be any number between -35 and 35. To express this mathematically, we can write it as a double inequality: - Determine the values that satisfy: - \(-35 < x < 35\)- This gives us the range of \(x\) values that solve the inequality.Thus, solving inequalities involves rewriting expressions so that they directly reveal a range of possible solutions.

Symbolic Equation Solving

Solving an equation symbolically means finding exact values of the variable that satisfy the equation using algebraic methods. Let’s look at solving \(|x| - 10 = 25\). The key is to isolate the absolute value expression \(|x|\) by adding 10 to both sides, simplifying the equation to \(|x| = 35\).The equation \(|x| = 35\) implies that the value inside the absolute value, \(x\), can be either 35 or -35 since both yield an absolute value of 35.Thus, solving symbolically involves:- Recognizing that absolute value equations model distance.- Using basic operations (in this case, addition) to isolate the absolute value.- Writing solutions for both the positive and negative scenarios, hence the solutions are \(x = 35\) and \(x = -35\). This symbolic approach ensures we identify all possible solutions.

Interval Notation

Interval notation is a concise way to describe a set of solutions, particularly for inequalities. Once we solved the inequality \(|x| < 35\), we found that \(x\) can take any value between -35 and 35. In interval notation, this range is expressed as:- \((-35, 35)\)- This notation indicates all numbers greater than -35 and less than 35 are solutions.A few key points about interval notation:- Parentheses \(()\) are used for inequalities that are "less than" or "greater than" (open interval), indicating that the endpoints are not included.- Square brackets \([]\) are used if the endpoints are included (closed interval), such as for "less than or equal to".Using interval notation makes it easier to communicate solution sets for inequalities quickly and effectively.