Problem 71 - Algebra 2

Question

Solve each equation. Check your answers. $$ \frac{1}{2}|3 x+2|-3=4 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 4 \) and \( x = -\frac{16}{3} \), but upon checking only \( x = 4 \) satisfies the original equation.

Step 1: Isolate the absolute value expression

First, add 3 to both sides of the equation to isolate the absolute value expression on one side. So, the equation becomes \( \frac{1}{2}|3x + 2| = 4 + 3 \).

Step 2: Simplify the equation

Next, simplify the right side of the equation by adding the numbers together: \( \frac{1}{2}|3x + 2| = 7 \).

Step 3: Remove the fraction

Multiply both sides of the equation by 2 to eliminate the fractional coefficient. You will get \( |3x + 2| = 14 \).

Step 4: Split into two equations

The absolute value equation can be split into two separate equations. This is because the expression inside the absolute value brackets can be either positive or negative: \( 3x + 2 = 14 \) and \( 3x + 2 = -14 \).

Step 5: Solve the first equation

Solve the first equation \( 3x + 2 = 14 \) by subtracting 2 from both sides, then dividing by 3, giving us \( x = \frac{14 - 2}{3} = 4 \).

Step 6: Solve the second equation

Solve the second equation \( 3x + 2 = -14 \) by subtracting 2 from both sides, then dividing by 3, giving us \( x = \frac{-14 - 2}{3} = -\frac{16}{3} \) or \( x = -\frac{16}{3} \).

Step 7: Check the solutions

Replace \( x \) with 4 in the original equation and verify that both sides of the equation are equal. Then, replace \( x \) with \( -\frac{16}{3} \) and check again. The solution that does not satisfy the original equation is discarded.

Key Concepts

Understanding Absolute Value Expressions

Absolute value expressions are pivotal in understanding how to deal with distance and magnitude in mathematics. In essence, the absolute value of a number is its distance from zero on the number line, disregarding any direction (positive or negative). For instance, both 3 and -3 have an absolute value of 3, denoted as \(|3| = 3\) and \(|-3| = 3\).

When dealing with absolute value expressions in equations, they convey a 'no matter the sign' scenario, which requires contemplation of both the positive and negative outcomes. Therefore, such expressions can sometimes lead to two potential solutions in the context of equation solving because both \(x = a\) and \(x = -a\) can result in the same absolute value \(|a|\).

Isolating the Variable

Isolating the variable is a foundational technique in solving equations, including those involving absolute value expressions. It means rearranging the equation so that the variable of interest stands alone on one side of the equation. This process usually involves performing algebraic operations such as adding, subtracting, multiplying, or dividing both sides by the same number, to maintain equality.

For the sample equation \(\frac{1}{2}|3x + 2| - 3 = 4\), you would start by adding 3 to both sides to isolate the absolute value expression. This methodical process simplifies the equation and preps it for further steps to determine the variable's value.

Splitting Absolute Value

Angled brackets indicate the presence of an absolute value, which can disguise either a positive or a negative scenario. Because of this dual nature, equation solving requires 'splitting' the absolute value into two separate equations to cover both cases. These twin equations strip the absolute value, leaving a 'plus' version and a 'minus' version.

In our example, once the absolute value of \(3x + 2\) is isolated and equals 14, we split the equation into \(3x + 2 = 14\) and \(3x + 2 = -14\). These separate equations represent the crux of determining all possible solutions for the variable \(x\).

Equation Solving Steps

The approach to solving an equation generally follows a logical sequence of steps that progress from simplifying to solving. After isolating the variable and managing any absolute value splits, you solve the resulting equations getting as close to the form \(x = \text{value}\) as possible. This often means carrying out further operations such as subtracting constants or dividing by coefficients.

For instance, after splitting the absolute value from our exercise, we continue by subtracting 2 from both sides and then dividing by 3 to fully isolate \(x\), following a systematic framework that reliably leads to the solution.

Checking Equation Solutions

Once potential solutions are obtained, verification is essential. This 'checking' step ensures that our solutions actually satisfy the original equation, considering the principle of absolute value which admits two possibilities. As a diligent method, substitute each solution back into the original equation and verify if the left-hand side equals the right-hand side.

An incorrect solution often arises from overlooking restrictions imposed by the absolute value. By conscientiously checking each solution, we secure our understanding of the solutions' validity in the specific context of the equation.