Problem 51 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |6 x+5|=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{5}{6}\).

Step 1: Understanding Absolute Value

The absolute value, denoted as \(|a|\), represents the distance of a number \(a\) from zero on the number line, which is always a non-negative number. This implies that the expression \(|6x + 5|\) must be greater than or equal to zero.

Step 2: Analyze the Equation

The equation given is \(|6x + 5| = 0\). For the absolute value of an expression to equal zero, the expression inside the absolute value must be zero. This is because the distance of zero from zero is zero.

Step 3: Solve for the Expression Inside the Absolute Value

Since \(|6x + 5| = 0\), set the expression inside the absolute value equal to zero: \[6x + 5 = 0\].

Step 4: Isolate the Variable

To find \(x\), subtract 5 from both sides of the equation to isolate the term containing \(x\):\[6x = -5\].

Step 5: Solve for x

Divide both sides by 6 to solve for \(x\):\[x = \frac{-5}{6}\].

Key Concepts

Solving Equations

Understanding how to solve equations is a fundamental skill in math. A typical equation consists of two expressions set equal to each other. In our case, it involves the equation \(|6x + 5| = 0\). To solve it, we need to find the value of the variable \(x\) that makes this equation true. For equations involving absolute values, the approach is slightly different. Absolute value equations reflect the distance of an expression from zero on the number line. Solving these may involve considering several cases, but for this particular equation, there's a simplifying factor: the absolute value is equal to zero. When the absolute value of an expression equals zero, as in \(|6x + 5| = 0\), it tells us that the expression within the absolute value is, in fact, zero itself. Therefore, we solve the simpler equation \(6x + 5 = 0\), leading us to the next stages in the solution.

Isolate the Variable

Before we can find the value of the variable, we need to focus on isolating it. The idea is to get \(x\) alone on one side of the equation. In our equation \(6x + 5 = 0\), the term \(6x\) is accompanied by a constant, 5. To isolate \(6x\), you need to manipulate the equation such that it stands alone. The first step is to subtract 5 from both sides. This gets rid of the constant on the left:\[6x = -5\]This act of manipulation, by moving terms across the equal sign while adjusting their signs accordingly, helps simplify the equation step by step until the variable is isolated.

Expression Inside Absolute Value

The reason absolute value expressions often need special attention is due to their properties. The expression inside the absolute value tells us what we are working with directly. Whenever you have an equation like \(|6x + 5| = 0\), it's crucial to examine what's inside the absolute value, which is \(6x + 5\) in this case.The absolute value represents the distance a number is from zero, which means unless the entire expression equals zero, it can't be negative or simply disregarded. In essence, if the result equals zero, like in our exercise, it's because the expression inside the absolute value is zero. Hence we solve \(6x + 5 = 0\).Once you figure out this core detail about the expression inside the absolute value, solving the equation becomes straightforward: it's treated like any regular linear equation once the absolute value braces are removed.