Problem 6 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ |6 x|=15 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\).

Step 1: Understand Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. This means \(|6x| = 15\) implies both \(6x = 15\) and \(6x = -15\). We'll solve for both cases.

Step 2: Solve the First Equation

First, solve the equation \(6x = 15\). Divide both sides by 6 to isolate \(x\).\[ x = \frac{15}{6} = \frac{5}{2} \]

Step 3: Solve the Second Equation

Next, solve the equation \(6x = -15\). Similarly, divide both sides by 6 to isolate \(x\).\[ x = \frac{-15}{6} = -\frac{5}{2} \]

Step 4: Verify the Solutions

Verify that both solutions satisfy the original equation. Substitute \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\) into \(|6x| = 15\):- For \(x = \frac{5}{2}\): \(6 \times \frac{5}{2} = 15\), \(|15| = 15\), true.- For \(x = -\frac{5}{2}\): \(6 \times -\frac{5}{2} = -15\), \(|-15| = 15\), true.

Key Concepts

Solve Equations

Solving equations is a vital part of algebra. It is the essential process of finding the unknown variable that satisfies the given mathematical statement. In the case of absolute value equations, you need to consider both the positive and negative scenarios because absolute values depict a number's distance from zero, bypassing any signs.
To solve an equation like \(|6x| = 15\), we begin by recognizing that there are two potential equations stemming from the absolute value:

• \(6x = 15\)
• \(6x = -15\)

These represent both scenarios due to the nature of absolute values. By tackling both, you ensure that you investigate every possible solution. For each equation, you isolate \(x\) by dividing both sides of the equation by 6. This step is crucial for finding the precise solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations without an equality sign. Understanding these is fundamental when solving equations, especially when they involve unknowns. Absolute value equations like \(|6x| = 15\) transform into separate algebraic expressions:

• First expression: \(6x = 15\)
• Second expression: \(6x = -15\)

These expressions are tackled individually, simplifying such that you solve for the unknown, \(x\). Algebra comes into play as you manage fractions and simplify them into recognizable, user-friendly solutions.

Solutions Verification

Once you've pinned down potential solutions, it's time to verify them. Verification ensures the accuracy of your answers, affirming they truly satisfy the original equation. For the equation \(|6x| = 15\), the verification involves plugging back the solutions into the absolute value equation:

• Substitute \(x = \frac{5}{2}\) back into \(|6x|\). Calculate \(6 \times \frac{5}{2} = 15\) which yields \(|15| = 15\), confirming correctness.
• Similarly, plug \(x = -\frac{5}{2}\) into \(|6x|\). Compute \(6 \times -\frac{5}{2} = -15\), resulting in \(|-15| = 15\), also correct.

By cross-checking your solutions, you make sure that the process was done flawlessly, with legitimate solutions that fit the original equation. Verification is the final safeguard against errors, providing confidence in your algebraic problem-solving skills.