Problem 10 - Intermediate Algebra

Question

Determine whether \(-3\) is a solution of the given equation or inequality. a. \(|x-1|=4\) b. \(|x-1|>4\) c. \(|x-1| \leq 4\) d. \(|5-x|=|x+12|\)

Step-by-Step Solution

Answer

a: Yes; b: No; c: Yes; d: No.

Step 1: Understanding Absolute Value Equations

For part (a), we need to check if substituting \(-3\) into the equation \(|x-1|=4\) results in a true statement. Absolute value represents the distance from zero on the number line, so we substitute \(-3\) for \(x\): \(|-3 - 1| = 4\). Simplifying gives \(|-4| = 4\), which is true.

Step 2: Checking Inequality Condition

For part (b), we substitute \(-3\) into the inequality: \(|x-1|>4\) becomes \(|-3 - 1| > 4\). This simplifies to \(4 > 4\), which is false.

Step 3: Evaluating Inequality for "Less Than or Equal To"

For part (c), substitute \(-3\) into the inequality \(|x-1| \leq 4\), resulting in \(|-3 - 1| \leq 4\). This gives \(4 \leq 4\), which is true.

Step 4: Solving Absolute Value Equation

For part (d), substitute \(-3\) into the equation \(|5-x|=|x+12|\). This becomes \(|5 - (-3)| = |-3 + 12|\) which simplifies to \(|8| = |9|\). This statement is false because 8 is not equal to 9.

Key Concepts

Understanding Absolute Value Equations

An absolute value equation involves an expression with absolute value symbols. Absolute value is the non-negative "distance" a number is from zero on the number line. In mathematical terms, the absolute value of a number \(x\) is written as \(|x|\). The key property of absolute values is:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

For the equation \(|x - 1| = 4\), we check if substituting \(-3\) for \(x\) makes the equation true. Substituting gives \(|-3 - 1| = 4\), leading to \(|-4| = 4\). Since the absolute value of \(-4\) is 4, this equation holds true. The solution satisfies the equation because the absolute value maintains equality regardless of the negative sign within the expression.

Exploring Inequalities Involving Absolute Values

Inequalities with absolute values express conditions where distances must be greater than, less than, or equal to a certain number. When substituting values into such inequalities, it's crucial to simplify and check the resulting statement for truth. For \(|x - 1| > 4\), replacing \(x\) with \(-3\) gives us: \(|-3 - 1| > 4\), simplifying to \(4 > 4\), which is not true. This means \(-3\) is not a solution for this inequality.In contrast, checking \(|x - 1| \leq 4\) with \(x = -3\) results in: \(|-3 - 1| \leq 4\), simplifying to \(4 \leq 4\), which is true. Thus, \(-3\) satisfies this condition.These explorations demonstrate how different types of inequalities (greater than versus less than or equal to) impact the evaluation of solutions.

Using Substitution Method in Equations

The substitution method is a powerful tool for testing if a specific number is a solution to an equation or inequality. It involves replacing the variable with a given number and simplifying the expression to check for correctness. This method helps confirm the validity of solutions.Consider the equation \(|5 - x| = |x + 12|\). With \(x = -3\), substituting gives: \(|5 - (-3)| = |-3 + 12|\), simplifying to \(|8| = |9|\). Here, the simplification shows \(8 = 9\), which is false, indicating \(-3\) does not satisfy this equation.Substitution is a straightforward way to determine whether a number meets the conditions set by an equation or inequality, providing a clear conclusion about its status as a solution.