Problem 27 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|2 x+3.6|=9.8\)

Step-by-Step Solution

Answer

The solutions are \(x = 3.1\) and \(x = -6.7\).

Step 1: Understand the Absolute Value Definition

The absolute value of a number is its distance from zero on a number line, without considering which direction from zero the number lies. Therefore, the equation \(|2x + 3.6| = 9.8\) means that either \(2x + 3.6 = 9.8\) or \(2x + 3.6 = -9.8\).

Step 2: Solve the First Equation

Start by solving the equation \(2x + 3.6 = 9.8\).Subtract 3.6 from both sides:\[2x = 9.8 - 3.6\]\[2x = 6.2\]Next, divide both sides by 2:\[x = \frac{6.2}{2}\]\[x = 3.1\].

Step 3: Solve the Second Equation

Now solve the equation \(2x + 3.6 = -9.8\).Subtract 3.6 from both sides:\[2x = -9.8 - 3.6\]\[2x = -13.4\]Next, divide both sides by 2:\[x = \frac{-13.4}{2}\]\[x = -6.7\].

Step 4: Compile the Solutions

The solutions to the equation are the values of \(x\) found in Steps 2 and 3. Therefore, the solutions are \(x = 3.1\) and \(x = -6.7\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. It is a fundamental concept in algebra and requires a systematic approach. In this exercise, we are dealing with an absolute value equation, \(|2x + 3.6| = 9.8\).When working with absolute value equations, it is important to remember that the expression inside the absolute value can equal either the positive or the negative of the value on the other side of the equation. This is because the absolute value represents the distance from zero, which is always positive or zero. Therefore, we break the problem down into two possible situations:

• The expression equals 9.8, that is: \(2x + 3.6 = 9.8\)
• The expression equals -9.8, that is: \(2x + 3.6 = -9.8\)

By solving these two separate linear equations, we find the solutions. This approach guarantees we consider all possible values that satisfy the original absolute value equation.

Algebraic Solutions

Algebraic solutions involve manipulating equations to isolate the variable of interest. This process often includes using arithmetic operations such as addition, subtraction, multiplication, and division.To solve the first equation, \(2x + 3.6 = 9.8\), we start by eliminating the constant term on one side. Subtract 3.6 from both sides to simplify:\[ 2x = 9.8 - 3.6 \]\[ 2x = 6.2 \]Next, divide both sides by 2 to solve for \(x\):\[ x = \frac{6.2}{2} \]\[ x = 3.1 \]For the second equation, \(2x + 3.6 = -9.8\), repeat the process. Subtract 3.6:\[ 2x = -9.8 - 3.6 \]\[ 2x = -13.4 \]Then, divide by 2:\[ x = \frac{-13.4}{2} \]\[ x = -6.7 \]These steps collectively give us the algebraic solutions \(x = 3.1\) and \(x = -6.7\). Careful attention to signs and arithmetic leads to a complete solution.

Distance from Zero

The absolute value of a number is its distance from zero on a number line. This concept is essential when solving absolute value equations such as \(|2x + 3.6| = 9.8\).Absolute values disregard whether a number is positive or negative, focusing only on how far it is from zero. Hence, solutions to absolute value equations consider two scenarios: when the inside expression is equal to the positive and negative of a given number.For example, in this exercise, the absolute value indicates that \(2x + 3.6\) is 9.8 units away from zero. This situation is depicted as two possible equations, \(2x + 3.6 = 9.8\) and \(2x + 3.6 = -9.8\), since both values have the same distance from zero on a number line.Understanding this fundamental idea helps in visualizing how absolute values operate and solving equations efficiently.