Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write the solution set of each equation. $$ |x+8|=6 $$

Step-by-Step Solution

Answer

The solution set is \(\{-2, -14\}\).

Step 1: Understanding Absolute Value

The expression \(|x + 8| = 6\) means that the distance of \(x + 8\) from zero is 6. Absolute value equations like this one have two possible cases.

Step 2: Set Up Two Equations

To find the solutions, we need to consider both cases:1. \(x + 8 = 6\)2. \(x + 8 = -6\)These equations correspond to the two possible values \(x + 8\) can take to have an absolute value of 6.

Step 3: Solve First Equation

For the equation \(x + 8 = 6\), subtract 8 from both sides:\[x + 8 - 8 = 6 - 8\]\[x = -2\]

Step 4: Solve Second Equation

For the equation \(x + 8 = -6\), subtract 8 from both sides:\[x + 8 - 8 = -6 - 8\]\[x = -14\]

Step 5: Write the Solution Set

The solutions to the equation \(|x + 8| = 6\) are the values of \(x\) found in steps 3 and 4. Therefore, the solution set is \(\{-2, -14\}\).

Key Concepts

Solving Absolute Value Equations

Absolute value equations often puzzle students at first glance because they involve understanding how distance works on a number line rather than simple algebraic manipulation. An absolute value equation like \(|x + 8| = 6\) asks, "What values make the expression inside the absolute value sign, \(x + 8\), equal to 6 units away from zero?"
This means there are two potential answers or solutions because distance is not directional:

• The expression can either equal +6 ("positive case")
• Or it can equal -6 ("negative case")

By setting up two separate equations from the absolute value equation, we begin to solve them independently:
1. \(x + 8 = 6\) 2. \(x + 8 = -6\)
This step lays the groundwork for identifying specific values for \(x\) that satisfy the original equation.

Solution Set

The solution set of an equation includes all solutions that satisfy the equation. When we solve an absolute value equation, such as \(|x + 8| = 6\), we find multiple solutions due to the nature of absolute values representing distance.
For our equation, we discovered solutions by solving:

• \(x + 8 = 6\), yielding \(x = -2\)
• \(x + 8 = -6\), yielding \(x = -14\)

Each solution is a value for \(x\) making \(x + 8\) exactly 6 units away from zero. So, the solution set for \(|x + 8| = 6\) is \(\{-2, -14\}\). It is simply a grouping or bracketed list of all valid solutions, often denoted using curly braces, indicating the full array of solutions the equation holds on a number line.

Distance from Zero

In mathematics, absolute value is essentially about measuring how far a number or expression is from zero, no matter in which direction. Absolute value equations like \(|x + 8| = 6\) boil down to finding numbers that have a specified distance from zero.
This concept is crucial as it brings a geometric perspective into algebra, transforming numeric solutions into points or distances along a number line. When solving \(|x + 8| = 6\):

• The expression can be at +6, meaning \(x + 8 = 6\)
• Or it can be at -6, meaning \(x + 8 = -6\)

For each scenario, you find an \(x\) value, translating distance (6 units away) into specific numeric solutions: \(-2\) and \(-14\). This description makes it easier to grasp where and why solutions lay on a number line in relation to zero.

Algebraic Equations

An algebraic equation is like a balance scale, where you must find the values of an unknown that keep the scale balanced. Equations encompass all problems where two sides equal each other, often including variables. To solve \(|x + 8| = 6\), we reframe it into two separate algebraic equations that can easily be manipulated:1. \(x + 8 = 6\)2. \(x + 8 = -6\)
To isolate \(x\), perform the same operation on both sides of each equation (in this case, subtracting 8):

• For \(x + 8 = 6\):\[x + 8 - 8 = 6 - 8\] simplifying to \(x = -2\)
• For \(x + 8 = -6\):\[x + 8 - 8 = -6 - 8\] simplifying to \(x = -14\)

These transformed equations might look simpler, but they’re powerful in finding solution sets for absolute value and other algebraic equations, illustrating balance and equivalence fundamentals.