Problem 70 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$7|3 x|+2=16$$

Step-by-Step Solution

Answer

The solutions are \(x = 2/3\) and \(x = -2/3\).

Step 1: Isolate the Absolute Value

Subtract 2 from both sides of the equation to isolate the absolute value on one side: \(7|3x|=16-2 \rightarrow 7|3x|=14\). Then divide each side by 7 to further isolate the absolute value: \( |3x| = 14/7 \rightarrow |3x| = 2\).

Step 2: Remove the Absolute Value

Removing the absolute value gives two equations: \(3x = 2\) and \(3x = -2\).

Step 3: Solve Each Equation

Division by 3 yields the solutions for x: \(x = 2/3\) for the first equation, and \(x = -2/3\) for the second equation.

Key Concepts

Solving Absolute Value

To effectively comprehend how to solve absolute value equations, one must first understand what an absolute value represents. The absolute value of a number is the distance of that number from zero on the number line, regardless of its direction. As such, it's always non-negative. When solving equations that involve absolute value, the principle hinges on knowing that the expression inside the absolute value brackets can have two possible values: one positive and one negative.

Therefore, in solving the equation like the one from our exercise, once the absolute value is isolated, you remove the absolute value symbol and construct two separate equations to resolve. For instance, if you have \(|3x| = 2\), you create two scenarios: \(3x = 2\) and \(3x = -2\). Each of these equations is then solved normally, yielding the possible solutions to the original absolute value equation.

Isolate Absolute Value

The process to isolate the absolute value in an equation is a pivotal step in finding the solution. To \'isolate\' means to get the absolute value expression by itself on one side of the equation, allowing you to apply the definition of absolute value effectively. The isolation process typically involves applying basic algebraic operations such as addition, subtraction, multiplication, and division.

In our example, \(7|3x| + 2 = 16\), we first subtract 2 from both sides to start the isolation process. Simplifying further, we then divide by 7, leading us to \(|3x| = 2\). It's essential to perform these steps one at a time to avoid any errors and ensure clarity in the process. Always check every step by substituting back into the original equation to confirm that the resulting expressions are accurate and viable.

Algebraic Equations

At the heart of solving these problems are algebraic equations, which are mathematical statements that express the equality of two algebraic expressions. These equations can include variables, numbers, and operations. The solution to an algebraic equation is the value(s) of the variable(s) that make the equation true.

When tackling algebraic equations, one must carefully apply the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is succinctly remembered by the acronym PEMDAS. In the context of absolute value equations, once the equation is reduced to its basic algebraic form, such as \(3x = 2\) or \(3x = -2\), we solve for the variable by applying inverse operations. For example, to solve for x, we divide both sides by 3, which is the inverse operation of multiplication, resulting in our solutions \(x = \frac{2}{3}\) and \(x = -\frac{2}{3}\).