Problem 99 - Intermediate Algebra

Question

Solve. $$ |3 x-4|=11 $$

Step-by-Step Solution

Answer

The solutions are \(x = 5\) and \(x = -\frac{7}{3}\).

Step 1: Understanding Absolute Value

An absolute value equation like \(|3x - 4| = 11\) means that the expression inside the absolute value, \(3x - 4\), can be equal to 11 or its opposite, -11.

Step 2: Set Up Two Equations

To solve \(|3x - 4| = 11\), we set up two separate equations: \(3x - 4 = 11\) and \(3x - 4 = -11\).

Step 3: Solve the First Equation

For the first equation, add 4 to both sides: \(3x = 15\). Then divide both sides by 3 to isolate \(x\): \(x = 5\).

Step 4: Solve the Second Equation

For the second equation, add 4 to both sides: \(3x = -7\). Then divide both sides by 3 to isolate \(x\): \(x = -\frac{7}{3}\).

Step 5: Check the Solutions

Verify both solutions by plugging them back into the original equation. For \(x = 5\), \(|3(5) - 4| = |15 - 4| = |11| = 11\), which is correct. For \(x = -\frac{7}{3}\), \(|3(-\frac{7}{3}) - 4| = |-7 - 4| = |-11| = 11\), which is also correct.

Key Concepts

Step by Step Solution

Breaking down an absolute value equation is simpler than it might seem at first. The given equation, \[|3x - 4| = 11\], involves understanding the nature of the absolute value.
The absolute value of a number is its distance from zero on the number line. This means the inside of the absolute value—the expression \(3x - 4\)—could be either a positive or a negative result that equals 11.
To tackle this equation, we create two distinct scenarios, or equations, from the initial setup. First, we consider \(3x - 4 = 11\). Solving this involves adding 4 to both sides to get \(3x = 15\) and then dividing by 3, leading us to \(x = 5\).
The second scenario is to consider the negative possibility: \(3x - 4 = -11\). Again, we add 4 to both sides, resulting in \(3x = -7\). Divide by 3 to isolate \(x\), arriving at \(x = -\frac{7}{3}\).
So, the step-by-step solving process gives us two potential solutions: \(x = 5\) and \(x = -\frac{7}{3}\).

Checking Solutions

Verifying the solutions in mathematical problems is crucial to ensure accuracy. Once solutions are found, we don't just accept them without a second thought. Let's check our solutions for the absolute value equation \[|3x - 4| = 11\].
First, we substitute \(x = 5\) back into the original equation:

• Calculate \(3(5) - 4\), which simplifies to \(15 - 4\).
• This equals 11, confirming \(|11| = 11\).

Next, test \(x = -\frac{7}{3}\):

• Insert \(-\frac{7}{3}\) to get \(3(-\frac{7}{3}) - 4\).
• This results in \(-7 - 4 = -11\).
• The absolute value, \|-11|, is also 11, affirming the equation holds true.

Both values satisfy the original equation, confirming they are correct solutions. Checking such steps ensures we didn't make any algebra mistakes.

Solving Linear Equations

Solving linear equations involves clear steps to isolate the variable in question. For our problem, each derived equation is linear, meaning it can form a straight line when graphed. Let's consider how to solve these effectively.
Given the equation, say \(3x - 4 = 11\):

• Start by undoing the subtraction. Add 4 to both sides, converting it into \(3x = 15\).
• Next, remove the coefficient of \(x\) by dividing both sides by 3, leading to \(x = 5\).

Similarly, handle \(3x - 4 = -11\):

• Add 4 to both sides, yielding \(3x = -7\).
• Divide each side by 3 to find \(x = -\frac{7}{3}\).

The essence of solving linear equations lies in reversing operations—undoing what has been done to the variable. This methodically distills down the problem to its simplest form, revealing the solution. Understanding this allows for solving more complex problems efficiently.