Problem 13 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ |4 x+7|=9 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute Values

The equation involves an absolute value, which means that we must consider both the positive and negative scenarios that satisfy the equation. So, we write: 1. \(|4x + 7| = 9\), which translates to two separate equations: - \(4x + 7 = 9\) - \(4x + 7 = -9\).

Step 2: Solving the First Equation

Solve the equation \(4x + 7 = 9\). Subtract 7 from both sides to get:\[ 4x = 9 - 7 \]Simplify the right side:\[ 4x = 2 \]Now, divide both sides by 4:\[ x = \frac{2}{4} \]Simplify the fraction:\[ x = \frac{1}{2} \].

Step 3: Solving the Second Equation

Now, solve the equation \(4x + 7 = -9\). Subtract 7 from both sides:\[ 4x = -9 - 7 \]Simplify the right side:\[ 4x = -16 \]Now, divide both sides by 4:\[ x = \frac{-16}{4} \]Simplify the fraction:\[ x = -4 \].

Step 4: Writing the Solution Set

Since the absolute value equation has two potential solutions, our final answer includes both solutions from the steps above:\[ x = \frac{1}{2} \quad \text{or} \quad x = -4 \].

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics, particularly when dealing with equations. The absolute value of a number refers to its distance from zero on the number line, regardless of its direction.
Therefore, the absolute value of any real number is always non-negative. For example, the absolute value of both -3 and 3 is 3, because both are 3 units away from zero.
When we solve equations that include absolute values, like \(|4x + 7| = 9\), we must consider two scenarios. One scenario is the expression inside the absolute value being equal to the positive result, and the other scenario is it being equal to the negative result.

• This dual scenario approach accounts for the two possible distances from zero that could satisfy the absolute value equation.
• Thus, the equation splits into two separate linear equations.

Solving Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised only to the power of one. In the context of an absolute value equation, once the equation is split into two linear equations, these equations can be solved independently.
For instance, based on our original problem \(|4x + 7| = 9\), it splits into two linear equations:

• \(4x + 7 = 9\)
• \(4x + 7 = -9\)

Solving each of these equations requires isolating the variable. This is typically achieved by performing operations to simplify the equation. Subtraction, addition, and division are applied to both sides of the equation equally to maintain balance. Once simplified, you reach a solution for \(x\) that solves the linear equation.
In our example:

• For \(4x + 7 = 9\): Subtracting 7 from both sides and dividing by 4 gives \(x = \frac{1}{2}\).
• For \(4x + 7 = -9\): Subtracting 7 from both sides and dividing by 4 gives \(x = -4\).

Determining the Solution Set

The solution set of an equation contains all the possible values that satisfy it. After working through an absolute value equation, it often results in two possible solutions, due to the nature of absolute values producing two possible cases.
For our equation \(|4x + 7| = 9\), the solution set is derived from solving both linear equations formed from considering the positive and negative cases – namely, \(4x + 7 = 9\) and \(4x + 7 = -9\).

• The first equation provides the solution \(x = \frac{1}{2}\).
• The second equation yields the solution \(x = -4\).

Therefore, the complete solution set, which includes all possible values for \(x\), is \(\{ \frac{1}{2}, -4 \}\). This means both values are valid solutions that make the original absolute value equation true.