Problem 6 - Intermediate Algebra

Question

For Problems \(1-16\), solve each equation. $$ |5 x-7|=14 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4.2\) and \(x = -1.4\).

Step 1: Understand the Absolute Value Equation

To solve an equation involving absolute values, understand that the expression inside the absolute value, \(|5x-7|\), can be equal to 14 or -14.

Step 2: Set Up Two Separate Equations

Since the absolute value expression can be positive or negative, set up two separate equations based on the expression: 1. \(5x - 7 = 14\)2. \(5x - 7 = -14\)

Step 3: Solve First Equation

For the first equation \(5x - 7 = 14\), add 7 to both sides to get \(5x = 21\). Then divide by 5 to solve for \(x\), giving \(x = \frac{21}{5}\) or \(x = 4.2\).

Step 4: Solve Second Equation

For the second equation \(5x - 7 = -14\), add 7 to both sides to get \(5x = -7\). Then divide by 5 to solve for \(x\), giving \(x = \frac{-7}{5}\) or \(x = -1.4\).

Step 5: Verify the Solutions

Check both solutions in the original equation to ensure they satisfy \(|5x - 7| = 14\). For \(x = 4.2\), \(|5 \times 4.2 - 7| = 14\). For \(x = -1.4\), \(|5 \times (-1.4) - 7| = 14\). Thus, both solutions are correct.

Key Concepts

Solving Absolute Value Equations

To solve an absolute value equation like \(|5x - 7| = 14\), you need to understand what absolute value means. Absolute value is the distance from zero on the number line, regardless of direction. This means when you see \(|5x - 7|\), it can equal either 14 or -14 because both are 14 units from zero.
To solve this type of equation, you'll break it into two separate equations.

• First equation: Set the inside of the absolute value equal to 14, which gives us \(5x - 7 = 14\).
• Second equation: Set the inside of the absolute value equal to -14, which gives us \(5x - 7 = -14\).

By setting up these two equations, you are considering both possible cases for the expression inside the absolute value. This is a standard technique in solving absolute value equations.

Step by Step Solution

Solving the absolute value equation is straightforward if you follow it step by step. Let's break down the process:

• Step 1: Solve First Equation. Start with \(5x - 7 = 14\). Add 7 to both sides to get \(5x = 21\). Then, divide both sides by 5, resulting in \(x = \frac{21}{5}\) or \(x = 4.2\).


• Step 2: Solve Second Equation. Now tackle \(5x - 7 = -14\). Similarly, add 7 to both sides, resulting in \(5x = -7\). Next, divide both sides by 5, leading to \(x = \frac{-7}{5}\) or \(x = -1.4\).

By treating each equation separately, you simplify the task of solving an absolute value equation. This step by step approach ensures no solution is overlooked.

Verification of Solutions

After finding solutions in absolute value equations, it's crucial to verify them. Verification involves substituting your solutions back into the original equation to ensure they work.
For our solutions, substitute back into \(|5x - 7| = 14\):

• Verify for \(x = 4.2\): Compute \(5 \times 4.2 - 7 = 21 - 7 = 14\). Absolute value is 14, which satisfies the equation.


• Verify for \(x = -1.4\): Calculate \(5 \times -1.4 - 7 = -7 - 7 = -14\). The absolute value of -14 is 14, hence it satisfies the equation.

Verification confirms that both \(x = 4.2\) and \(x = -1.4\) are accurate solutions, ensuring the problem is fully solved. It's a vital step to verify the accuracy and see the logic behind the abstract math.