Question

\(|3 x+2|=5\)

Short answer

The solutions of the equation \(|3x + 2| = 5\) are \(x = 1\) and \(x = -7/3\) or \(x = -2.33\)

Step-by-step solution

Break down the equation

Given the equation \(|3x + 2| = 5\), break it down into two separate equations: \(3x + 2 = 5\) and \(3x + 2 = -5\)

Solve the first equation

Solve the first equation \(3x + 2 = 5\) for the variable x. To get x by itself, subtract 2 from both sides, which gives you \(3x = 3\). Then, divide both sides of the equation by 3 to find that \(x = 1\) is a solution to the equation.

Solve the second equation

Then solve the second equation \(3x + 2 = -5\) for the variable x. Similarly, subtract 2 from both sides, which gives you \(3x = -7\). Dividing both sides by 3 renders that \(x = -7/3\) or \(x = -2.33\) is another solution to the equation.

Key concepts

Understanding Absolute Value Equations

When solving absolute value equations, it's essential to comprehend the absolute value function. The absolute value of a number represents its distance from zero on a number line, regardless of direction. Therefore, the absolute value is always non-negative.

For the equation \( |3x + 2| = 5 \), the absolute value sign dictates that the expression inside can equal either 5 or -5. This dual possibility leads to the creation of two separate equations: \( 3x + 2 = 5 \) and \( 3x + 2 = -5 \). Each linear equation is then solved individually to find all possible solutions for x.

To ensure a clear understanding, remember that the absolute value equation can be thought of as a balance scale. The value within the absolute value signs must balance out to match the number on the other side of the equation, whether that number is positive or negative.

Tackling Linear Equations

Linear equations, such as \( 3x + 2 = 5 \) or \( 3x + 2 = -5 \), are the simplest form of equations to solve in algebra. They contain variables raised to the first power and yield a straight line when graphed.

In our exercise, solving the linear equations requires isolating the variable x. This is done by performing inverse operations. For \( 3x + 2 = 5 \), subtract 2 from both sides to undo the addition, leaving \( 3x = 3 \). Then, divide by 3, the coefficient of x, to find the solution \( x = 1 \).

Similarly, for \( 3x + 2 = -5 \) subtract 2 from both sides for \( 3x = -7 \)). Then divide by 3, leading to the solution \( x = -2.33 \). Always perform the same operations to both sides of the equation to maintain balance and arrive at the correct solution.

Achieving Algebraic Solutions

To derive algebraic solutions, knowledge of algebraic manipulation is key. This means employing operations such as addition, subtraction, multiplication, and division to both sides of an equation to solve for an unknown variable.

In the context of the given exercise, two solutions emerge from the algebraic process— namely \( x = 1 \) and \( x = -2.33 \). These solutions are obtained by systematically reversing the operations that have been applied to x. The strategy is straightforward: simplify, isolate, and solve.

When dealing with absolute value equations, always ensure you derive two separate solutions due to the nature of absolute values. Moreover, algebraic solutions not only offer the specific answers but also enhance understanding of the underlying mathematical principles.