Problem 26 - Precalculus: Functions and Graphs

Question

Solve the equation. \(2|5 x+2|-1=5\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{5}\) and \(x = -1\).

Step 1: Isolate the Absolute Value Expression

First, we need to isolate the absolute value expression on one side of the equation. Start by adding 1 to both sides of the equation:\[2|5x + 2| - 1 + 1 = 5 + 1\]This simplifies to:\[2|5x + 2| = 6\]

Step 2: Divide by 2 to Further Simplify

Next, divide both sides of the equation by 2 to simplify the absolute value expression:\[\frac{2|5x + 2|}{2} = \frac{6}{2}\]This gives:\[|5x + 2| = 3\]

Step 3: Solve the Two Absolute Value Equations

We now solve the equation \(|5x + 2| = 3\) by considering the two possible cases.Case 1: \(5x + 2 = 3\)Subtract 2 from both sides:\[5x = 1\]Divide by 5:\[x = \frac{1}{5}\]Case 2: \(5x + 2 = -3\)Subtract 2 from both sides:\[5x = -5\]Divide by 5:\[x = -1\]

Step 4: Provide the Solutions

From the two cases considered in Step 3, we have two solutions to the original equation: \(x = \frac{1}{5}\) and \(x = -1\). Remember, both need to be validated in the context of the absolute value equation, ensuring they satisfy the initial condition. Both solutions do satisfy the condition here.

Key Concepts

Equation Solving

When tackling absolute value equations, the primary goal is to isolate the absolute value expression. This allows us to deal with the equation more effectively. Absolute value equations often result in more than one solution because of the nature of absolute values. They represent distance from zero, making the expression inside the absolute value positive or negative.
In solving an absolute value equation like \(2|5x + 2| - 1 = 5\), the first step is adding or subtracting terms to get the absolute value expression by itself on one side of the equation. This is crucial because it simplifies the form of the equation, making it easier to solve. After isolating the absolute value expression, we divide or multiply to further simplify before considering the cases for the equation.
It's important to always check that the solutions obtained satisfy the original equation. So, once you've identified potential solutions, plug them back into the original equation to ensure they hold true.

Step by Step Solutions

Breaking down the problem into smaller, clear steps can significantly help in understanding and solving complex equations like those involving absolute values.
Here is a breakdown of the usual steps you follow in solving such equations:

• **Isolate the Expression:** Start by manipulating the equation so that the absolute value term is isolated. This may involve adding, subtracting, multiplying, or dividing both sides.
• **Simplify Further:** Once isolated, it's often necessary to further simplify by dividing or multiplying, as needed.
• **Consider All Cases:** For example, with \(|5x + 2| = 3\), you need to consider both \(5x + 2 = 3\) and \(5x + 2 = -3\) to fully solve the equation.
• **Solve and Validate:** Solve these individual equations, arriving at possible solutions, and check back in the original equation to ensure that the solutions are correct.

This kind of structured approach is invaluable for securing reliable solutions with confidence.

Precalculus Problems

In precalculus, understanding absolute value equations is vital, as it's a stepping stone to more complex problems in calculus and advanced mathematics. Here, you deal with functions and expressions that represent real-world situations.
Understanding how to handle absolute value gives students insight into solving more complicated functions. It's not just about solving for \(x\), but also understanding what the solutions represent in a problem.
Precalculus problems often require you to work with transformations, shifts, and scaling of graphs, and absolute value expressions often require similar analytical approaches. Make sure to practice different absolute value problems to get comfortable with switching perspectives – such as looking at an equation from an algebraic standpoint and a graphical one. This flexibility will greatly aid in tackling more complex math problems later.