Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ 3|x+1|-2=-11 $$

Step-by-Step Solution

Answer

No solution, since absolute values can't be negative.

Step 1: Isolate the Absolute Value

First, we need to isolate the absolute value on one side of the equation. We start with the equation:\[3|x+1|-2=-11 \]Add \(2\) to both sides of the equation:\[3|x+1| = -11 + 2 \] Which simplifies to:\[3|x+1| = -9\]

Step 2: Divide to Fully Isolate the Absolute Value

Next, divide both sides of the equation by 3 to solve for \(|x+1|\):\[|x+1| = \frac{-9}{3} \]Simplifying gives:\[|x+1| = -3 \]

Step 3: Analyze the Absolute Value Situation

An absolute value expression is always non-negative, meaning \(|x+1|\) cannot be negative. Since \(|x+1|\) equals \(-3\), this situation is impossible, indicating there is no real value for \(x\) that satisfies this equation.

Key Concepts

Solving Absolute Value Equations

Solving equations involving absolute values can seem tricky at first. But, once you get the hang of the process, it becomes much more manageable. Absolute value equations like the one given can have two possible interpretations because the absolute value of a number is its distance from zero on the number line, always resulting in a non-negative value.

When dealing with an absolute value equation such as \(|x + 1| = a\), you need to consider both possible cases:

• Case 1: \(x + 1 = a\)
• Case 2: \(x + 1 = -a\)

Both equations must be solved separately. In our problem, however, the equation \(|x+1| = -3\) results in a nonsensical scenario since absolute values cannot be negative. Hence, no solution exists.

Remember these important points:

• The expressions within the absolute value should be isolated first before attempting to solve the equation.
• Check whether the absolute value expression is set equal to a negative number, as this usually implies no solution exists.

Understanding Algebra

Algebra often forms the foundation for solving mathematical equations and can help you express real-world situations using mathematical expressions. When working with absolute value equations, algebra is the tool that allows us to manipulate expressions to isolate terms efficiently.

Let's explore key components of managing algebraic equations:

• Transform equations through addition, subtraction, multiplication, and division.
• Keep equations balanced by performing the same operation on both sides. This maintains equality.
• Use distributive properties and simplifications to make equations simpler to solve.

By mastering these algebraic techniques, you're equipped to handle more complex equations, making algebra an essential skill.When solving our equation \(3|x+1| - 2 = -11\), the aim was first to isolate \(|x+1|\). We did this by applying inverse operations, which is a fundamental algebraic process. Isolating expressions often involves multiple steps, requiring a strong grasp of algebraic manipulation.

Isolating Terms

The objective of isolating terms in equations is to simplify them step-by-step until the unknown variable is clearly identified. This technique is key in both basic algebra and more complex calculus problems. Here’s why isolating terms is important:

• It systematically breaks down the equation, making it easier to manage.
• Ensures you can clearly see the relationship between variables.
• Facilitates the process of evaluating whether solutions are possible, as seen in our case with absolute values.

To isolate terms efficiently, follow these strategies:

• Perform inverse operations to both sides of the equation.
• Simplify algebraic expressions at each step to make subsequent steps easier.
• Rearrange terms logically to focus on the variable of interest.

In our example of isolating \(3|x+1|-2=-11\), adding \(2\) to both sides and then dividing by \(3\) helped isolate the term \(|x+1|\). Recognizing when terms have been fully isolated is crucial, especially when dealing with absolute values, as it helps identify infeasible solutions early on.