Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(|x-10| \geq 3\) (a) \(x=13\) (b) \(x=-1\) (c) \(x=14\) (d) \(x=9\)

Step-by-Step Solution

Answer

Hence, none of the given values \(x=13, -1, 14, 9\) are solutions to the inequality \(|x - 10| \geq 3\).

Step 1: Solve the absolute inequality

An absolute inequality \(|x - a| \geq b\), can be divided into two separate inequalities, \(x - a \geq b\) and \(-(x - a) \geq b\). Solving for \(x\) in both inequalities will give the range of values for which the absolute inequality holds. For \(|x - 10| \geq 3\), this becomes \(x - 10 \geq 3\) and \(-(x - 10) \geq 3\), resulting in \(x \geq 13\) and \(x \leq 7\).

Step 2: Test given values for \(x\)

Now, substitute the given values for \(x\) into the solved inequality to test whether they are solutions. (a) For \(x=13\), \(13 \geq 13\) and \(13 \leq 7\), the first condition holds but the second does not, so \(x=13\) is not a solution. (b) For \(x=-1\), \(-1 \geq 13\) and \(-1 \leq 7\), both conditions do not hold, so \(x=-1\) is not a solution. (c) For \(x=14\), \(14 \geq 13\) and \(14 \leq 7\), the first condition holds but the second does not, so \(x=14\) is not a solution. (d) For \(x=9\), \(9 \geq 13\) and \(9 \leq 7\), both conditions do not hold, so \(x=9\) is not a solution.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. This means it's always a non-negative number. For example, for any number \(x\), the absolute value is denoted by \(|x|\). A helpful way to think about this is as a measure of "size" without worrying about the sign.

When dealing with absolute value inequalities like \(|x - 10| \geq 3\), you must understand that it expresses all \(x\) values whose distance from 10 is at least 3 units. To solve, you break it into two separate inequalities:

• The expression \(x - 10\) should be greater than or equal to 3.
• Alternatively, \(-(x - 10)\) should be greater than or equal to 3, capturing the opposite scenario by flipping the expression sign.

This method ensures you consider distances on both sides of 10, covering the full set of possible solutions.

Solution of Inequality

In solving inequalities, the approach is about finding the set of parameters that satisfy a condition instead of a specific number. For \(|x - 10| \geq 3\), two inequalities emerge:

• \(x - 10 \geq 3\) which implies \(x \geq 13\)
• \(10 - x \geq 3\) leading to \(x \leq 7\)

Both represent essential conditions of distance, reflecting positions on opposite sides of 10.

An interval solution for this inequality involves finding values for \(x\) that fulfill either condition, hence the union of \(x \leq 7\) and \(x \geq 13\). Verifying given values requires testing each number to see if it meets one or both conditions, ensuring correct and comprehensive understanding. In this scenario, none of the given values \(x = 13, -1, 14, 9\) strictly satisfy the inequality because they fall within the distance too close to 10.

Algebraic Expressions

Understanding algebraic expressions involves grasping how they function as representations of relationships between quantities. In inequalities, you often rearrange expressions to identify solution sets.

For instance, in the inequality \(|x - 10| \geq 3\), we first recognize the expression \(x - 10\). Rearranging is a crucial skill that allows the break down of absolute value into linear inequalities, assisting in identifying possible solutions. Simplifying each algebraic expression assists in visualizing and categorizing values that satisfy each condition.

• Position \(x\) in such a manner that fulfills either \(x \geq 13\) or \(x \leq 7\)
  
• Use properties of addition, subtraction, multiplication, and division interchangeably, while preserving the inequality's direction unless dealing with a negative multiplier.
  
• The result will match translated values per the initial expression, conveying solution intervals effectively.
  

Mastering algebraic expressions grants a handle over solving complex problems, aiding in swiftly pinpointing viable solutions.