Problem 15 - Algebra for College Students

Question

For Problems \(15-54\), solve each equation and inequality. \(|x-1|=8\)

Step-by-Step Solution

Answer

The solutions are \(x = 9\) and \(x = -7\).

Step 1: Understand Absolute Value Equations

The equation \(|x - 1| = 8\) involves an absolute value. In general, for an equation \(|A| = B\), where \(B > 0\), the solutions are \(A = B\) and \(A = -B\). This means the expression inside the absolute value can be either positive or negative and still result in the absolute value of \(8\).

Step 2: Remove the Absolute Value

Apply the property of absolute values to remove the absolute value bars and set up two separate equations: \(x - 1 = 8\) and \(x - 1 = -8\).

Step 3: Solve the First Equation

For the equation \(x - 1 = 8\), add \(1\) to both sides to isolate \(x\): \[\begin{align*}x - 1 &= 8 \x &= 8 + 1 \x &= 9\end{align*}\]

Step 4: Solve the Second Equation

For the equation \(x - 1 = -8\), add \(1\) to both sides: \[\begin{align*}x - 1 &= -8 \x &= -8 + 1 \x &= -7\end{align*}\]

Step 5: Write the Solution Set

Combine the solutions from both equations. The solution set for \(|x - 1| = 8\) is \(x = 9\) or \(x = -7\). This means the absolute value equation has two solutions.

Key Concepts

Solving Equations

Understanding how to solve equations is fundamental in mathematics. An equation is a statement that asserts the equality of two expressions. The main objective of solving an equation is to find the unknown value that satisfies the equation. Here's a simple process:

• First, identify what you need to solve. In our case, it's the absolute value equation \(|x - 1| = 8\).
• Next, apply any rules or properties relevant to the type of equation. For absolute values, this involves setting up the two scenarios: one where the inside expression equals the positive of the other side and one where it equals the negative.
• Finally, use algebraic methods to isolate the unknown variable and solve the equations individually.

Always remember, when solving equations, check your solutions by plugging them back into the original equation to ensure they hold true.

Absolute Value Properties

Absolute value is a key mathematical concept that measures how far a number is from zero, regardless of direction. The absolute value of a number is always non-negative. The symbol for absolute value is two vertical bars: \(|x|\).An essential property of absolute value comes into play when solving equations. For any real number \(A\) and any positive number \(B\), the equation \(|A| = B\) leads to two possibilities:

• The expression inside the bars is equal to \(B\).
• The expression inside the bars is equal to \(-B\).

This understanding allows us to transform absolute value equations into two simpler linear ones. This concept is crucial when dealing with the absolute value in equations, helping us solve them effectively by considering both potential solutions.

Equation Solving Steps

Once you know how to handle absolute values, solving the equation \(|x - 1| = 8\) becomes straightforward. Let's break down the steps:First, recognize the equation as an absolute value equation. This triggers the need to remove the absolute value bars by setting up two separate equations:

• \(x - 1 = 8\)
• \(x - 1 = -8\)

Proceed by solving each equation one at a time:

• Add 1 to both sides of \(x - 1 = 8\) to find \(x = 9\).
• Similarly, for \(x - 1 = -8\), add 1 to both sides to arrive at \(x = -7\).

Finally, gather the solutions. The solution set includes both values, meaning the equation \(|x - 1| = 8\) is satisfied when \(x\) is 9 or -7. These final steps illustrate not only the solution process but also reinforce the dual nature of solving absolute value equations.