Problem 59 - Algebra for College Students

Question

Solve each equation and inequality by inspection. \(|5 x-2|=0\)

Step-by-Step Solution

Answer

\(x = \frac{2}{5}\)

Step 1: Identify the Absolute Value Equation

The given equation is \(|5x - 2| = 0\). The property of absolute value states that for \(|a| = 0\), \(a\) must be 0. Therefore, we set the inside of the absolute value equal to 0: \(5x - 2 = 0\).

Step 2: Solve for x

Now solve the equation \(5x - 2 = 0\). Add 2 to both sides to isolate the term with \(x\): \(5x = 2\).

Step 3: Isolate x

Divide both sides by 5 to solve for \(x\): \(x = \frac{2}{5}\).

Step 4: Verify the Solution

Substitute \(x = \frac{2}{5}\) back into the original equation to ensure it satisfies \(|5x - 2| = 0\):- Substitute: \(5\left(\frac{2}{5}\right) - 2 = 0\).- Simplify: \(2 - 2 = 0\).- Since both sides equal 0, the solution is correct.

Key Concepts

Solving Equations

When faced with an equation, your goal is to find out what value the variable must take for the equation to hold true. In this case, our equation involves an absolute value: \(|5x - 2| = 0\). Absolute value represents the distance of a number from zero, regardless of direction on the number line. Therefore, the absolute value of a number is zero only if the number itself is zero. This is important. It means that when \(|a| = 0\), then \(a = 0\). So, to solve the equation, set the inside of the absolute value expression equal to zero: \(5x - 2 = 0\). This simplifies the problem to a basic linear equation. To solve this, add 2 to both sides:

• \(5x - 2 + 2 = 0 + 2\)
• \(5x = 2\)

Next, divide by 5 to isolate \(x\):

• \(x = \frac{2}{5}\)

Solving Inequalities

While our original exercise involves solving an equation, understanding how inequalities work is crucial, especially with absolute values. If you have an absolute value inequality, such as:\(|5x - 2| < 3\) or \(|5x - 2| > 3\),it can be split into two separate cases:

• For \(|5x - 2| < 3\), split it into \(5x - 2 < 3\) and \(5x - 2 > -3\).
• For \(|5x - 2| > 3\), split it into \(5x - 2 > 3\) or \(5x - 2 < -3\).

The solution will consist of a range of values that satisfy the inequality. Solve each inequality like a regular equation, isolating \(x\) for the solution. With inequalities, always pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers, as it reverses the inequality.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions in equations or inequalities to isolate the variable. Let's revisit how we achieved this with the original \(5x - 2 = 0\): We performed two main operations:

• Adding 2 to both sides to eliminate the constant term next to \(5x\): \(5x - 2 + 2 = 0 + 2\)
• Dividing each side by 5 to solve for \(x\): \(x = \frac{2}{5}\)

These steps show how specific operations can simplify equations and isolate variables for solutions. Similarly, for more complex equations involving fractions or multiple terms, you will use techniques like combining like terms or factoring. Becoming proficient in algebraic manipulation provides a foundation for tackling more advanced equations and improving your problem-solving skills.