Problem 65 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|2 x-1|=5$$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -2\).

Step 1: Formation of Equations

From \(|2x-1|=5\), two equations can be formed: \(2x-1=5\) and \(2x-1=-5\).

Step 2: Solving for x in \(2x-1=5\)

Add 1 to both sides of the equation \(2x-1=5\) which gives \(2x = 6\). Then divide both sides by 2 to isolate x, giving \(x = 3\).

Step 3: Solving for x in \(2x-1=-5\)

Add 1 to both sides of the equation \(2x-1=-5\) which gives \(2x = -4\). Then divide both sides by 2 to isolate x, giving \(x = -2\).

Key Concepts

Solving Absolute Value Equations

When we talk about solving absolute value equations, we're dealing with equations that include expressions inside absolute value bars, denoted as \(|...|\). Absolute value represents the distance a number is from zero on the number line, which means it's always a non-negative number. Therefore, an equation involving absolute value can have up to two possible solutions.
To solve an absolute value equation, such as \(|2x-1| = 5\), you begin by setting up two separate equations. This accounts for the fact that what's inside the absolute value can either be equal to a positive value (the distance itself) or its negative counterpart. For our example, you form the equations:

• \(2x - 1 = 5\)
• \(2x - 1 = -5\)

By solving these equations individually, you find all possible values of \(x\) that satisfy the original absolute value equation. This step is crucial, as it ensures no solution is overlooked.

Absolute Value

The absolute value of a number is the numerical value without regard to its sign. If \(x\) is any real number, the absolute value \(|x|\) is defined as:

• \(x\), if \(x \geq 0\)
• -x, if \(x < 0\)

It's important to understand that the absolute value is fundamentally about distance. For example, both 3 and -3 have an absolute value of 3, because each is three units from zero on the number line.
The property of absolute value to turn negative expressions to positive is what allows equations like \(|2x-1| = 5\) to possibly have more than one solution. This inherent nature makes it a powerful tool in algebra for highlighting multiple scenarios, and it requires careful consideration and handling in equations.

Algebraic Equations

Algebraic equations are statements of equality involving algebraic expressions. These expressions can include variables, constants, and arithmetic operations. The focus of solving algebraic equations is to find the values of the variables that make the statement true.
For instance, given the equation \(2x - 1 = 5\), we apply operations to isolate \(x\). This gives us insight into what value \(x\) would be so that, when substituted back into the equation, both sides are equal. The operations used are generally reversible actions like adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number.
Clearing the equation can often result in simple forms like \(x = 3\), but complex algebraic equations may require advanced methods, such as factoring, quadratic formula, or substitution, to solve.

Solution of Equations

The solution of an equation is the set of values for the variables that satisfy the equation, making it true. When dealing with absolute value equations as in \(|2x-1| = 5\), finding the solution involves calculating the outcomes of multiple scenarios due to the nature of absolute values.
In our discussed example, solving both \(2x - 1 = 5\) and \(2x - 1 = -5\) gives the set of solutions as \(x = 3\) and \(x = -2\). These solutions indicate that if you replace \(x\) with either 3 or -2 in the original equation \(|2x-1| = 5\), it holds true, fulfilling the condition of the absolute value equation.
It's essential to verify each solution by substituting back into the original equation, thus ensuring no missteps in simplification or assumptions during the solving process. Understanding the solutions also involves recognizing when an equation has no solution, which can happen if the absolute value is set equal to a negative number, an impossibility under standard definitions.