Problem 32 - Algebra 1

Question

Solve the equation. $$|-8|+x=-3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -11\).

Step 1: Simplify Absolute Value

First, simplify the absolute value of -8. The absolute value of -8 is 8, so the equation becomes \(8 + x = -3\).

Step 2: Solve for x

Next, isolate \(x\) by subtracting 8 from both sides of the equation to get \(x = -3 - 8\).

Step 3: Simplify the Right Side

Finally, simplify the right side of the equation to get \(x = -11\).

Key Concepts

Understanding Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of the direction. To put it simply, it is the 'numerical value' of a number without considering its sign. For instance, the absolute value of both \( -8 \) and \( 8 \) is \( 8 \), because both points are 8 units away from zero on the number line.

When you encounter an absolute value in an equation, such as \( |-8| \), the first step is always to simplify it by removing the absolute value and replacing it with its non-negative value, leading to \( 8 \) in this case. This is an essential step because it simplifies the equation and prepares it for further steps to isolate the variable.

Isolating the Variable

To solve an equation for a variable means to isolate the variable on one side of the equation. In the given problem, isolating \( x \) is necessary to find its value. This typically involves performing basic arithmetic operations such as addition, subtraction, multiplication, and division, while following the rule of performing the same operation on both sides of the equation to maintain equality.

After simplifying the absolute value, the next step is to subtract \( 8 \) from both sides of the equation \( 8 + x = -3 \) to isolate \( x \) which renders \( x = -3 - 8 \). The subtraction operation effectively 'moves' the \( 8 \) to the other side, thus isolating \( x \) on the left side of the equal sign.

Simplifying Equations

Simplifying equations is about combining like terms and reducing expressions to their simplest form to make them easier to solve. After you have isolated the variable, as we did with \( x \) in this example, the following step is to simplify the expression on the other side.

In the problem \( x = -3 - 8 \), both terms on the right-hand side are numbers, which means they can be added together (keeping the sign in mind). By simplifying \( -3 - 8 \) we obtain \( -11 \) which is the solution to the equation. The simplification process is crucial as it reveals the solution in its most understandable form.