Problem 40 - Algebra 2

Question

Solve each equation. Check your solutions. \(|3 x-7|-5=-3\)

Step-by-Step Solution

Answer

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

Step 1: Isolate the Absolute Value

To solve the equation \(|3x - 7| - 5 = -3\), begin by isolating the absolute value expression. We do this by adding 5 to both sides of the equation.\(|3x - 7| - 5 + 5 = -3 + 5\)This simplifies to:\(|3x - 7| = 2\)

Step 2: Solve the Absolute Value Equation

The equation \(|3x - 7| = 2\) can split into two separate equations:1. \(3x - 7 = 2\)2. \(3x - 7 = -2\)We'll solve each of these equations separately.

Step 3: Solve the First Equation

Take the first equation:\(3x - 7 = 2\).Add 7 to both sides:\(3x = 2 + 7\)\(3x = 9\)Now, divide both sides by 3:\(x = \frac{9}{3}\)\(x = 3\)

Step 4: Solve the Second Equation

Now, solve the second equation \(3x - 7 = -2\):Add 7 to both sides:\(3x = -2 + 7\)\(3x = 5\)Divide both sides by 3:\(x = \frac{5}{3}\)

Step 5: Check the Solutions

Let's check both solutions in the original equation.For \(x = 3\):\(|3(3) - 7| - 5 = -3\)\(|9 - 7| - 5 = -3\)\(|2| - 5 = -3\)\(2 - 5 = -3\)Since both sides equal, \(x = 3\) is correct.For \(x = \frac{5}{3}\):\(|3\left(\frac{5}{3}\right) - 7| - 5 = -3\)\(|5 - 7| - 5 = -3\)\(|-2| - 5 = -3\)\(2 - 5 = -3\)Since both sides equal, \(x = \frac{5}{3}\) is correct.

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. This means that whether a number is positive or negative, its absolute value is always non-negative. For instance, the absolute value of both 5 and -5 is 5. The absolute value is denoted by the vertical bars around the number or expression, such as \(|x|\).

When solving absolute value equations like \(|3x - 7| = 2\), you break it into two possible scenarios because the expression inside the bars, \(3x - 7\), could either be 2 or -2.

• First scenario: \(3x - 7 = 2\)
• Second scenario: \(3x - 7 = -2\)

This method ensures that all possible solutions for the expression in the absolute value are accounted for.
Before diving into solving, it's a good practice to isolate the absolute value expression. This often involves moving terms outside the absolute value symbol to the other side of the equation, just like in the given problem where \(-5\) was moved to the other side to make the equation simpler.

Solving Linear Equations Arising from Absolute Value

After isolating the absolute value, you are usually left with linear equations like \(3x - 7 = 2\) and \(3x - 7 = -2\). Linear equations are equations of the first degree, meaning they involve no exponents greater than one on the variable, and typically take the form \(ax + b = c\). Solving these requires finding the value of the variable that makes the equation true.

Let's briefly review how to solve such linear equations:

• Start by isolating the variable term on one side. This usually involves adding or subtracting numbers from both sides.
• Next, divide or multiply to solve for the variable itself.

For example, in \(3x - 7 = 2\) you would add 7 to get \(3x = 9\), and then divide by 3, finding \(x = 3\). Similarly, solving \(3x - 7 = -2\), you would add 7 to get \(3x = 5\) and divide to find \(x = \frac{5}{3}\).

These simple yet effective steps allow you to handle any linear equation you might encounter when breaking down absolute value equations.

Checking Solutions for Accuracy

Checking solutions is a crucial step in solving equations to ensure that the obtained solutions truly satisfy the original equation. After finding possible values for the variable, substitute each back into the original equation to verify their correctness. Let's see how this applies practically:

• Substitute the solution back into the equation: Put each found value in place of the variable in the original absolute value equation.
• Check if both sides of the equation are equal: Perform the operations to ensure that the left side equals the right side.

For example, substituting \(x = 3\) back into the original problem \(|3(3) - 7| - 5 = -3\), simplifies to \(2 - 5 = -3\), which holds true. Similarly, checking \(x = \frac{5}{3}\) also results in a true statement, confirming it as a valid solution.

This verification process helps spot errors by ensuring all potential solutions work in the context of the original problem, offering a foolproof way to confirm accuracy.