Problem 40 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ \left|\frac{1}{9} x+4\right|+25=25 $$

Step-by-Step Solution

Answer

The solution is \( x = -36 \).

Step 1: Simplify the Equation

The given equation is \( \left| \frac{1}{9}x + 4 \right| + 25 = 25 \). Begin by subtracting 25 from both sides to isolate the absolute value expression. \[ \left| \frac{1}{9}x + 4 \right| + 25 - 25 = 25 - 25 \]This simplifies to: \[ \left| \frac{1}{9}x + 4 \right| = 0 \]

Step 2: Solve for the Absolute Value

The equation \( \left| \frac{1}{9}x + 4 \right| = 0 \) means that the expression inside the absolute value must also be zero. Therefore, we set the expression equal to zero: \[ \frac{1}{9}x + 4 = 0 \]

Step 3: Isolate x in the Equation

Subtract 4 from both sides to isolate the term with x:\[ \frac{1}{9}x = -4 \]Next, multiply both sides by 9 to solve for \( x \): \[ x = -4 \times 9 \]\[ x = -36 \]

Step 4: Verify the Solution

Substitute \( x = -36 \) back into the original absolute value equation to verify the solution.Calculate \( \frac{1}{9}(-36) + 4 \):\[ \frac{1}{9}(-36) + 4 = -4 + 4 = 0 \]Since the absolute value of 0 is 0, the original equation holds:\[ \left| 0 \right| + 25 = 25 \] So, the solution \( x = -36 \) is correct.

Key Concepts

Solving Equations

In the realm of mathematics, solving equations is a fundamental skill, especially when it comes to dealing with absolute value equations. An equation's solution involves finding the value of the unknown variable that makes the equation true. In our exercise, we started with an absolute value equation:

• \( \left| \frac{1}{9}x + 4 \right| + 25 = 25 \)

This task begins with simplifying the equation. Simplification is crucial because it sets the stage for isolating other components, often the variable itself. By subtracting 25 from both sides of the equation, we get:

• \( \left| \frac{1}{9}x + 4 \right| = 0 \)

This simplification directly leads to the next step, which is addressing the behavior of absolute values. When the absolute value equals zero, it implies that the expression inside must also be zero.

• The result is the simple statement \( \frac{1}{9}x + 4 = 0 \).

The journey within solving equations is a step-by-step voyage. Each step is key to unveiling the variable's value.

Isolation of Variables

The isolation of variables is a powerful technique in solving equations. It involves manipulating the equation in such a way that we progressively "isolate" the variable we want to solve for. Moving forward from the simplified equation \( \frac{1}{9}x + 4 = 0 \), we need to zero in on \( x \). First, we remove the constant term:

• Subtract 4 from both sides to yield \( \frac{1}{9}x = -4 \).

Once the variable term is isolated, the next step is to solve for \( x \) by eliminating the fraction:

• Multiply both sides by 9 to counteract the \( \frac{1}{9} \).
• This results in \( x = -36 \).

By isolating the variable, every transformation of the equation is aimed at getting closer to revealing the value of \( x \). This approach not only makes the variable "solvable" but also ensures clarity in the logical progression of algebraic manipulation.

Verification of Solutions

Verification of solutions is an essential final step in solving equations. It ensures the accuracy of the solution by substituting it back into the original equation to check consistency.
In our example, the solution obtained was \( x = -36 \). Now, verify by substituting back into the original setup:

• Calculate \( \frac{1}{9}(-36) + 4 \) to confirm the solution.
• This simplifies to \( -4 + 4 = 0 \).

Once we ascertain that the left side matches the right side of the equation, we establish the truth of our solution:

• \( \left| 0 \right| + 25 = 25 \), which proves correct.

Verification not only provides peace of mind but also bolsters confidence in the solution process. This step validates our initial steps and transformations and reassures us that the manipulations performed brought us to a correct and logical outcome.