91Ó°ÊÓ

Solving equations is a fundamental skill in algebra. The goal is to find the value of the variable that makes the equation true. Starting with an equation like \|x-3| = 11\, the approach is to isolate the variable step by step.

To isolate the variable \( x \), we first need to understand what the equation tells us. Here, \|x-3| = 11\ means the distance between \( x-3 \) and 0 is 11. Distance in absolute value means there can be two scenarios—one positive and one negative.

For the positive case, set \( x-3 = 11 \). For the negative case, set \( x-3 = -11 \). Then, solve each scenario individually:

• Add 3 to both sides in the first case: \( x-3 = 11 \rightarrow x = 14 \)
• Add 3 to both sides in the second case: \( x-3 = -11 \rightarrow x = -8 \)

Both scenarios give us potential solutions, so the solutions are \( x = 14 \) and \( x = -8 \).

The absolute value of a number is the distance between that number and 0 on a number line. It is always non-negative. For any real number \( a \), \|a| >= 0\.

When dealing with absolute value equations, we use the property that \|a| = b\ implies \a = b\ or \a = -b\. This property helps us break down the problem into two separate linear equations.

In our example, \|x-3| = 11\, the expression inside the absolute value, \( x-3 \), can be either 11 or -11. This leads to two cases:

• Case 1: \( x-3 = 11 \)
• Case 2: \( x-3 = -11 \)

Solving each case individually gives us the full set of solutions for the absolute value equation.

Linear equations are equations where the highest power of the variable is 1. These equations form a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is \( ax + b = c \).

In our example, both cases result in simple linear equations. For instance, \( x-3 = 11 \) and \( x-3 = -11 \) are both linear. We solve them in the same way:

• For \( x-3 = 11 \), add 3 to isolate \( x \): \( x = 14 \)
• For \( x-3 = -11 \), add 3 as well: \( x = -8 \)

Each step keeps the equation balanced and helps us systematically find the solution. Understanding linear equations simplifies solving more complex problems like those involving absolute values.