Problem 11 - Algebra and Trigonometry

Question

Solve the equation. $$ 4-|3 x+6|=1 $$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the equation. We have:\[4 - |3x + 6| = 1\].Subtract 4 from both sides to get:\[-|3x + 6| = 1 - 4\].This simplifies to:\[-|3x + 6| = -3\].Multiply both sides by -1 to obtain:\[|3x + 6| = 3\].Now the absolute value expression is isolated.

Step 2: Remove the Absolute Value

The equation \(|3x + 6| = 3\) means that the expression inside the absolute value can be either 3 or -3. Thus, we have two equations:\[3x + 6 = 3\]and\[3x + 6 = -3\].We will solve each equation separately.

Step 3: Solve the First Equation

Solve the equation\[3x + 6 = 3\].Subtract 6 from both sides to obtain:\[3x = 3 - 6\],which simplifies to:\[3x = -3\].Divide both sides by 3:\[x = \frac{-3}{3} = -1\].

Step 4: Solve the Second Equation

Now solve the equation:\[3x + 6 = -3\].Subtract 6 from both sides:\[3x = -3 - 6\],which simplifies to:\[3x = -9\].Divide both sides by 3:\[x = \frac{-9}{3} = -3\].

Step 5: Verify the Solutions

We found two solutions: \(x = -1\) and \(x = -3\). Substitute them back into the original equation to verify:1. For \(x = -1\):\(4 - |3(-1) + 6| = 4 - |3| = 4 - 3 = 1\), which is true.2. For \(x = -3\):\(4 - |3(-3) + 6| = 4 - |-3| = 4 - 3 = 1\), which is also true.Both values satisfy the original equation, confirming they are correct.

Key Concepts

Solving Equations

When tackling an equation like \(4 - |3x + 6| = 1\), the first step involves solving for the variable within the absolute value expression. The first part deals with getting the absolute value expression on one side of the equation by itself.
To do this, subtract 4 from both sides:

• Start with: \(4 - |3x + 6| = 1\)
• Subtract 4: \(-|3x + 6| = -3\)
• Multiply by -1 to clear the negative: \(|3x + 6| = 3\)

The central goal here is to isolate the absolute value part of the equation to make it easier to solve. Thereafter, you move to the next phase: removing the absolute value, which gives you two different equations to solve.
After isolating the absolute value, consider that \(|3x + 6| = 3\) implies two scenarios:

• The expression inside the absolute value is equal to 3
• Or it’s equal to -3

Algebraic Expressions

In algebra, absolute value symbols split the equation into two potential solutions because they can change a negative result into a positive one. Hence in \(|3x + 6| = 3\), we actually create two separate linear equations to solve:

• \(3x + 6 = 3\)
• \(3x + 6 = -3\)

By setting the algorithm within the absolute value both equal to 3 and -3, you can discover the potential solutions for \(x\). Let's look into solving both linear equations in detail.

For the equation \(3x + 6 = 3\): Subtract 6 from both sides to find \(3x = -3\). Dividing through by 3 gives \(x = -1\).
For the second scenario \(3x + 6 = -3\): Subtract 6 from both sides to yield \(3x = -9\). Dividing both sides by 3 results in \(x = -3\).

This process of breaking down an absolute value expression into two linear equations is crucial. It allows you to see both outcomes the absolute value could potentially hide.

Verification of Solutions

Once potential solutions \(x = -1\) and \(x = -3\) have been found, it's key to verify them with the original equation to ensure they satisfy it. Verification is a way of double-checking that no mistakes were made while solving.To verify, substitute each solution back into the initial equation:

• For \(x = -1\): Calculate \(4 - |3(-1) + 6|\). Inside, you get \(4 - |3|\), which results in \(4 - 3 = 1\). This concludes that \(x = -1\) is correct, as it holds true to the original equation.
• For \(x = -3\): Calculate \(4 - |3(-3) + 6|\). This simplifies within to \(4 - |-3|\), giving \(4 - 3 = 1\). Thus \(x = -3\) checks out as well.

Verifying solutions is an integral part of solving any equation to ensure its accuracy, by re-substituting your solutions you confirm their validity. This step ensures the mathematical process is correctly and thoroughly executed.