Problem 82 - Algebra 2

Question

Solve each equation. \(5|3 x-4|=x+1\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{2}\) and \(x = \frac{19}{16}\).

Step 1: Understand the Absolute Value Equation

The equation given is \(5|3x-4|=x+1\). This equation involves an absolute value, which means that \(3x-4\) can be either positive or negative. Therefore, we will solve two separate equations: one where \(3x-4\) is positive and one where it is negative.

Step 2: Solve the Equation when \(3x-4\) is Positive

When \(3x-4\) is positive, the absolute value function does not affect its sign, so the equation becomes \(5(3x-4) = x + 1\). Expand and solve for \(x\):\[ 15x - 20 = x + 1 \]Subtract \(x\) from both sides:\[ 14x - 20 = 1 \]Add 20 to both sides:\[ 14x = 21 \]Divide by 14:\[ x = \frac{21}{14} = \frac{3}{2} \]

Step 3: Solve the Equation when \(3x-4\) is Negative

When \(3x-4\) is negative, the absolute value function changes its sign, so the equation becomes \(5(-3x+4) = x + 1\). Expand and solve for \(x\):\[ -15x + 20 = x + 1 \]Add \(15x\) to both sides:\[ 20 = 16x + 1 \]Subtract 1 from both sides:\[ 19 = 16x \]Divide by 16:\[ x = \frac{19}{16} \]

Step 4: Verify the Solutions

Check both solutions in the context of the absolute value. For \(x = \frac{3}{2}\): \[ 3x - 4 = 3\left(\frac{3}{2}\right) - 4 = \frac{9}{2} - 4 = \frac{1}{2} \] Since \(\frac{1}{2}\) is positive, this solution is valid.For \(x = \frac{19}{16}\):\[ 3x - 4 = 3\left(\frac{19}{16}\right) - 4 = \frac{57}{16} - \frac{64}{16} = -\frac{7}{16} \] Since \(-\frac{7}{16}\) is negative, this solution is valid as well.

Key Concepts

Solving Absolute Value Equations

Solving an absolute value equation involves understanding what the absolute value represents. The absolute value of a number is always its distance from zero on the number line, which means it's always non-negative.
In the equation \(5|3x-4|=x+1\), due to the absolute value, \(|3x-4|\) can either be \(3x-4\) itself (if \(3x-4\) is positive) or \(-(3x-4)\) (if \(3x-4\) is negative). To find the solution, you need to consider both possibilities:

• Case 1: \(3x-4 \geq 0\), which gives \(5(3x-4) = x + 1\).
• Case 2: \(3x-4 < 0\), which gives \(5(-3x+4) = x + 1\).

After formulating these two equations, solve them individually to find the possible solutions for \(x\). Getting both solutions helps in verifying which, if not all, satisfy the absolute condition.

Utilizing Algebraic Techniques

Algebraic techniques are essential when solving equations involving absolute values. For the equation scenarios derived from the given absolute value problem, we apply basic manipulation techniques such as distribution, addition, subtraction, and division.
For instance, in solving \(5(3x-4)=x+1\), distribute the 5:

• First, expand to get \(15x - 20 = x + 1\).
• Next, perform operations such as subtraction, \(15x - x = 1 + 20\), to simplify.
• Finally, simplify to \(14x = 21\) and divide to find \(x = \frac{21}{14} = \frac{3}{2}\).

Each step ensures you're systematically isolating \(x\) to find the solution easily. These techniques are repeated for both scenarios, checking consistency with the absolute value condition.

Understanding Piecewise Functions

When dealing with absolute value equations, the concept of piecewise functions naturally comes into play. A piecewise function breaks down the domain of a function into "pieces" and applies different formulas to each piece.
In our equation \(5|3x-4|=x+1\), the absolute value \(|3x-4|\) behaves differently based on whether the expression inside is non-negative or negative:

• For \(3x-4 \geq 0\), the function is simply \(3x-4\).
• For \(3x-4 < 0\), the function switches to \(-(3x-4)\) or \(4-3x\).

Solving these separately ensures all possible solutions derived cater to the full range of the function's behavior. Understanding piecewise functions thus becomes crucial in retrieving all valid solutions for any absolute value equation.