Problem 63 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x-2|=7$$

Step-by-Step Solution

Answer

The solution to the equation \(|x-2|=7\) are \(x=9\) and \(x=-5\).

Step 1: Write Out the Two Possible Equations

The equation with an absolute value can be written as two separate equations. Hence, the equation \(|x-2|=7\) can be broken down into \(x-2=7\) and \(x-2=-7\).

Step 2: Solve the First Equation

In the equation \(x-2 = 7\), add '2' to each side of the equation to isolate 'x'. When you add '2' to the right hand side of the equation, we get \(x = 7 + 2\). This simplifies to \(x = 9\). Thus, the first solution is \(x=9\).

Step 3: Solve the Second Equation

Similarly for the equation \(x-2 = -7\), add '2' to each side of the equation to isolate 'x'. When you add '2' to the right hand side of the equation, we get \(x = -7 + 2\). This simplifies to \(x = -5\). Hence, the second solution is \(x = -5\).

Key Concepts

Absolute Value

Understanding the absolute value is crucial when solving equations that involve this concept. The absolute value of a number refers to its distance from zero on the number line, regardless of direction. It's denoted by two vertical bars, for example, \(|x|\). The absolute value of a number is always non-negative. When solving absolute value equations, we consider both the positive and the negative scenarios because if \(|x| = a\), then \(x\) could be \(a\) or \(-a\) since both would be \(a\) units away from zero on the number line. The equation \(|x-2|=7\) reveals that the quantity \(x-2\) is 7 units away from zero, so \(x-2\) can be 7 or -7.

Algebraic Equations

Algebraic equations are mathematical statements indicating that two expressions are equal. They contain variables, coefficients, and constants, and solving them involves finding the value of the variables that make the equation true.
For instance, \(x-2=7\) and \(x-2=-7\) are algebraic equations derived from the absolute value equation \(|x-2|=7\). The goal is to find the value of \(x\) that satisfies both conditions. Algebraic equations can describe a wide range of real-world situations and are fundamental in various fields of study.

Isolating Variables

Isolating the variable is a method used in solving equations where we manipulate the equation to get the variable by itself on one side of the equation. The aim is to find the value of this variable. To isolate the variable, one can perform various operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
In the exercise, we isolate \(x\) by adding \(2\) to both sides of each equation. This step results in \(x=9\) from the equation \(x-2=7\) and \(x=-5\) from the equation \(x-2=-7\). These operations ensure that we maintain the equality of both sides while finding the value of \(x\) that satisfies each equation.

Equation Solutions

The solutions to an equation are the values of the variables that make the equation true. In the context of absolute value equations, there are often two solutions due to the nature of absolute value, representing two possible distances from zero on the number line.
In the given problem, once the variable is isolated, we identify the solutions \(x=9\) and \(x=-5\), which satisfy the original equation \(|x-2|=7\). It is important to verify these solutions by substituting them back into the original equation to confirm their validity. This step ensures that no extraneous solutions have been included, which can occasionally arise during the process of solving algebraic equations.