Problem 20 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ \left|\frac{3}{5} x+2\right|-\frac{1}{2}=4 $$

Step-by-Step Solution

Answer

Solutions: \( x = \frac{25}{6} \) and \( x = -\frac{65}{6} \).

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. To do this, add \(\frac{1}{2}\) to both sides:\[\left|\frac{3}{5} x+2\right| = 4 + \frac{1}{2}\]This simplifies to:\[\left|\frac{3}{5} x+2\right| = \frac{9}{2}\]

Step 2: Remove the Absolute Value

The equation \(\left| A \right| = B\) implies that \( A = B \) or \( A = -B \). Therefore, we need to solve for \( \frac{3}{5}x + 2 = \frac{9}{2} \) and \( \frac{3}{5}x + 2 = -\frac{9}{2} \).

Step 3: Solve the First Case

Solve the equation \( \frac{3}{5}x + 2 = \frac{9}{2} \). Begin by subtracting 2:\[\frac{3}{5}x = \frac{9}{2} - 2\]Convert 2 to have a common denominator of 2:\[\frac{3}{5}x = \frac{9}{2} - \frac{4}{2} = \frac{5}{2}\]Then multiply both sides by \(\frac{5}{3}\):\[x = \frac{5}{2} \cdot \frac{5}{3} = \frac{25}{6}\]

Step 4: Solve the Second Case

Solve the equation \( \frac{3}{5}x + 2 = -\frac{9}{2} \). Begin by subtracting 2:\[\frac{3}{5}x = -\frac{9}{2} - 2\]Convert 2 to have a common denominator of 2:\[\frac{3}{5}x = -\frac{9}{2} - \frac{4}{2} = -\frac{13}{2}\]Then multiply both sides by \(\frac{5}{3}\):\[x = -\frac{13}{2} \cdot \frac{5}{3} = -\frac{65}{6}\]

Step 5: Conclusion

The solutions to the equation are the two values we solved for, so the solutions are \( x = \frac{25}{6} \) and \( x = -\frac{65}{6} \).

Key Concepts

Solving Absolute Value Equations

Solving absolute value equations involves a unique approach because of the nature of absolute values. Absolute value represents the distance from zero on a number line and can be thought of as making numbers positive. Therefore, equations involving absolute values require special handling.

Understanding the principle is key:

• Start by isolating the absolute value expression. In most cases, this means getting the absolute value on one side of the equation all by itself.
• Equations of the form \( |A| = B \) imply two possible scenarios: \( A = B \) or \( A = -B \).

Solving these separate equations provides the full set of potential solutions. This systematic approach allows us to investigate both the positive and negative scenarios that absolute value encompasses.

Algebraic Expressions

Algebraic expressions form the building blocks of equations and often appear inside absolute value brackets. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols.

In our example, the expression within the absolute value is \( \frac{3}{5}x + 2 \). Here:

• \( \frac{3}{5}x \) represents the variable portion multiplied by a fraction, which requires careful handling when solving.
• The +2 is a constant that adjusts the value of the expression as a whole.

Dealing with algebraic expressions involves operations to combine or simplify them, allowing for the isolation or resolution within equations.

Equation Isolation

Isolating equations is a crucial step in solving any mathematical problem because it simplifies the process. This involves manipulating the equation to have the variable or the expression you care about alone on one side.

Consider the initial equation \[ \left|\frac{3}{5} x + 2\right| - \frac{1}{2} = 4 \] We start by isolating the absolute value term. Add \( \frac{1}{2} \) to both sides to remove the subtraction, giving: \[ \left|\frac{3}{5} x + 2\right| = \frac{9}{2} \] This step is paramount because it sets the stage for removing the absolute value brackets and solving the simplified equation.

Common Denominator

The common denominator is a strategy used when dealing with fractions, especially when they need to be added or subtracted. It simplifies varying denominators to a single uniform value, making operations straightforward.

During equation solving, you often encounter fractions, as with\[ \frac{3}{5}x = \frac{9}{2} - \frac{4}{2} = \frac{5}{2} \] Here:

• The \( \frac{4}{2} \) adjustment requires making sure that both fractions share a common denominator.
• The denominator being the same allows direct subtraction, resulting in \( \frac{5}{2} \).

This technique is commonly used to simplify each side of an equation or to prepare fractions for further operations such as multiplication or division.