Question

Solve the equation. $$|x+8|=9$$

Short answer

The solutions for the equation are \(x = 1\) and \(x = -17\).

Step-by-step solution

Set up two equations

An absolute value equation is essentially two separate equations. This results from the fact that it's the sum of two cases: when \(x + 8\) is positive making \(x + 8 = 9\) and when \(x + 8\) is negative, making \(-(x + 8) = 9\). This gives us two equations to solve: \(x + 8 = 9\) and \(-x - 8 = 9\).

Solve the first equation

Solve \(x + 8 = 9\) for 'x'. To do so, subtract 8 from both sides, resulting in \(x = 9 - 8\) which simplifies to \(x = 1\).

Solve the second equation

Solve \(-x - 8 = 9\) for 'x'. Begin by adding 8 to both sides, resulting in \(-x = 9 + 8\), followed by simplifying to \(-x = 17\). Finally, multiply both sides by -1, to isolate 'x', yielding \(x = -17\).

Key concepts

Algebraic Equations

Algebraic equations are the cornerstone of algebra and form the basis of higher mathematics. These equations are statements that assert the equality of two expressions and often contain one or more unknown variables. Solving an algebraic equation means finding the value of the variables that make the equation true.

For instance, in the given exercise, the equation to solve was \(|x+8|=9\). Here, \(x\) is the unknown variable. The process of solving this or any algebraic equation involves a series of steps that simplify the equation to find that unknown variable's possible values.

Absolute Value

The absolute value of a number refers to its distance from zero on a number line, regardless of direction. Symbolized by two vertical lines | |, the absolute value is always non-negative. For example, the absolute value of both 3 and -3 is 3.

In terms of equations, absolute value can create a scenario where there are two possibilities to consider: when the inside of the absolute value (the expression) is positive or zero, and when it is negative. Therefore, when you solve an equation involving absolute values, you need to consider these separate cases for a comprehensive solution, as seen in the exercise where the equation \(|x+8|=9\) leads to two possible equations to solve.

Equation Solutions

Solving equations entails finding the value(s) of the variable(s) that satisfy the equation. Some equations have a single solution, while others, such as absolute value equations, may have multiple solutions. The goal is to isolate the variable on one side of the equation to determine its value.

In our exercise, there are two solutions to the absolute value equation: \(x = 1\) and \(x = -17\). These solutions arise from considering both the positive and negative scenarios of the expression within the absolute value. It's important to check each potential solution in the original equation to ensure it is valid.

Step-by-Step Problem-Solving

Step-by-step problem-solving is a systematic approach to understanding and addressing a question. It helps to break down complex problems into manageable steps, ensuring no part of the problem is overlooked and allowing a clear pathway to the solution.

In solving the absolute value equation \(|x+8|=9\), we used a three-step process. First, we set up two separate equations based on the absolute value concept. Next, we solved each equation individually, following algebraic principles—subtraction or addition of terms, followed by division or multiplication if necessary. Lastly, we obtained the solution: two numbers, \(x = 1\) and \(x = -17\), using methodical approaches, which are the roots of the given equation. This structured method aids in reducing mistakes and ensures a thorough comprehension of each step involved in reaching the solution.