Problem 15 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ 4-|3 x+6|=1 $$

Step-by-Step Solution

Answer

The solutions are \(x = -1\) and \(x = -3\).

Step 1: Simplify the Equation

Start by isolating the absolute value on one side of the equation. Subtract 3 from both sides of the equation:\[4 - |3x + 6| = 1 \Rightarrow |3x + 6| = 3.\]

Step 2: Consider Positive and Negative Scenarios

Recall that the absolute value equation \(|a| = b\) implies two scenarios: \(a = b\) and \(a = -b\). Therefore:\[3x + 6 = 3 \quad \text{or} \quad 3x + 6 = -3.\]

Step 3: Solve for x in the Positive Scenario

Consider the positive scenario \(3x + 6 = 3\). Subtract 6 from both sides:\[3x = 3 - 6 \3x = -3.\]Divide by 3 to solve for \(x\):\[x = -1.\]

Step 4: Solve for x in the Negative Scenario

Now consider the negative scenario \(3x + 6 = -3\). Subtract 6 from both sides:\[3x = -3 - 6 \3x = -9.\]Divide by 3 to solve for \(x\):\[x = -3.\]

Step 5: Verify the Solutions

Substitute each solution back into the original equation to verify:- For \(x = -1\): \(|3(-1) + 6| = |3| = 3\), satisfies \(4 - 3 = 1\).- For \(x = -3\): \(|3(-3) + 6| = |-3| = 3\), satisfies \(4 - 3 = 1\).Both solutions are verified.

Key Concepts

Solving Equations

To solve absolute value equations, we first need to isolate the absolute value expression on one side of the equation. This means modifying the equation so that the terms involving the absolute value are by themselves on one side. We can achieve this by using simple arithmetic operations: addition or subtraction.

• For example, in the given equation \(4 - |3x + 6| = 1\), we start by moving terms around to isolate the absolute value: \(|3x + 6| = 3\).
• Once isolated, we can proceed by considering different scenarios that absolute values can represent, namely the positive and negative cases of the expression inside the absolute value.

This step sets the stage for breaking down the equation into multiple manageable parts.

Positive and Negative Scenarios

In absolute value equations, you will often encounter two scenarios to consider for the solution. This arises because the absolute value of a number could make the inner expression either positive or negative. If you have an equation of the form \(|a| = b\), there are two possible equations you must solve:

• \(a = b\), representing the situation where the expression inside is unaffected by absolute value.
• \(a = -b\), accounting for the fact that a negative expression inside the absolute value becomes positive.

For instance, solving \(|3x + 6| = 3\), we develop two potential equations:

• \(3x + 6 = 3\)
• \(3x + 6 = -3\)

This strategic breakdown simplifies our equation into alternatives that can be solved individually.

Verifying Solutions

After deriving potential solutions from your absolute value equation, it is critical to verify them. Verification ensures your solutions satisfy the original equation, considering any transformations applied during solving. Here's how you verify:

• Substitute each potential solution back into the original equation.
• Check if both sides of the original equation remain equal when this substitution is made.

For example, with solutions \(x = -1\) and \(x = -3\), substituting them back into the equation \(4 - |3x + 6| = 1\) confirms:

• For \(x = -1\): Substitution gives \(|-3 + 6| = 3\), leading to \(1\).
• For \(x = -3\): Substitution results in \(||-9 + 6|| = 3\), also yielding \(1\).

This step is essential to confirm that all mathematical steps lead back to an accurate representation of the given problem.

Algebraic Manipulation

Mastering algebraic manipulation is key to solving equations efficiently and accurately. This skill involves employing operations such as addition, subtraction, multiplication, and division to transform and simplify equations. In the context of absolute value equations, follow these steps:

• Simplify both sides of the equation wherever possible.
• Effectively isolate the absolute value expression before considering positive and negative scenarios, as shown in \(4 - |3x + 6| = 1\), leading to \(|3x + 6| = 3\).
• Carefully handle each equation derived from the absolute scenarios using arithmetic operations.

Breaking down equations and systematically applying algebraic transformations help in moving step-by-step towards the solution. This approach encourages error-free manipulation and contributes to the verification process by ensuring each step is logical and accurate.