Problem 9 - Algebra and Trigonometry

Question

Solve the equation. $$ |4 x+7|=9 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -4\).

Step 1: Understand the Absolute Value Equation

The equation given is \(|4x + 7| = 9\). This means the expression inside the absolute value can equal 9 or -9.

Step 2: Set Up Two Separate Equations

Since \(|A| = B\) implies \(A = B\) or \(A = -B\), apply this to \(|4x + 7| = 9\). This gives us two equations: \(4x + 7 = 9\) and \(4x + 7 = -9\).

Step 3: Solve the First Equation

Solve for \(x\) in the equation \(4x + 7 = 9\). Subtract 7 from both sides: \(4x = 9 - 7\). Simplifying gives \(4x = 2\). Divide both sides by 4 to find \(x\): \(x = \frac{2}{4} = \frac{1}{2}\).

Step 4: Solve the Second Equation

Now solve for \(x\) in the equation \(4x + 7 = -9\). Subtract 7 from both sides: \(4x = -9 - 7\). This simplifies to \(4x = -16\). Divide both sides by 4 to find \(x\): \(x = \frac{-16}{4} = -4\).

Step 5: Write the Solution Set

The solutions from the two equations are \(x = \frac{1}{2}\) and \(x = -4\). Combine these results into a solution set: \(x \in \left\{ \frac{1}{2}, -4 \right\}\).

Key Concepts

Solving Absolute Value Equations

Absolute value equations are a type of equation where the main expression is within absolute value bars, like \( |4x + 7| = 9 \). The absolute value of a number is its distance from zero on the number line, so it is always positive or zero.

Here, solving the equation involves breaking it down into two separate cases. This is because the expression inside the absolute value, \(4x + 7\), can be equal to both 9 and -9:

• If \(4x + 7 = 9\), then by simple algebra, we find \(x = \frac{1}{2}\).
• On the other hand, if \(4x + 7 = -9\), solving gives \(x = -4\).

These are two linear equations we solve independently. Remember to always consider both the positive and negative scenarios when solving absolute value equations.
Breaking down the absolute value equation into two separate equations is the most crucial step in solving these types of algebraic problems. Always check your solutions to ensure they satisfy the original equation.

Understanding Solution Sets

Once we have solved for \(x\) in the individual equations derived from the absolute value equation, we need to compile these solutions into something called a solution set.

The solution set is essentially a list of all possible values that satisfy the original equation. In our case:

• We found \(x = \frac{1}{2}\) as a solution from the first equation \(4x + 7 = 9\).
• We also discovered \(x = -4\) from solving the second equation \(4x + 7 = -9\).

Thus, the solution set for \( |4x + 7| = 9 \) is \( x \in \left\{ \frac{1}{2}, -4 \right\} \). This notation means that \(x\) can either be \(\frac{1}{2}\) or \(-4\).

Creating a solution set allows for an organized way to present all possible answers, making it clear and concise for anyone reviewing the solution.

Working with Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as \( +, -, \, \times, \div \)). In solving absolute value equations, like \( |4x + 7| = 9 \), understanding how to manipulate these expressions is key. Here, the expression inside the absolute value is \(4x + 7\).

When working with algebraic expressions, you often need to:

• Perform arithmetic operations: This includes adding, subtracting, multiplying, or dividing.
• Understand and apply distribution laws and properties like combining like terms.
• Simplify expressions to isolate the variable, in this case, \(x\).

In our exercise, solving \(4x + 7 = 9\) involved:

1. Subtracting 7 from both sides to get \(4x = 2\).
2. Dividing each side by 4 to solve for \(x\), resulting in \(x = \frac{1}{2}\).

Similarly, solving \(4x + 7 = -9\) also required basic algebraic manipulation. Familiarity with algebraic expressions is crucial for solving equations effectively and finding accurate solutions.