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Chapter 8: Problem 39

Solve. $$ 7|q|-2=9 $$

Short Answer

Expert verified
The solutions are \( q = \frac{11}{7} \) and \( q = -\frac{11}{7} \).

Step by step solution

01

Isolate the Absolute Value

Add 2 to both sides of the equation to isolate the absolute value term.\[ 7|q| - 2 + 2 = 9 + 2 \] This simplifies to:\[ 7|q| = 11 \]
02

Solve for the Absolute Value

Divide both sides by 7 to solve for the absolute value.\[ |q| = \frac{11}{7} \]
03

Create Two Equations

Since \(|q| = \frac{11}{7}\), we have two cases: \( q = \frac{11}{7} \) or \( q = -\frac{11}{7} \)
04

State the Solutions

The solutions to the equation are: \( q = \frac{11}{7} \) and \( q = -\frac{11}{7} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The absolute value of a number refers to its distance from zero on the number line, regardless of direction. In other words, it is always a non-negative number. The absolute value is denoted by two vertical lines around the number or variable, such as \(|q|\). For example, \(|3| = 3\) and \(|-3| = 3\). The absolute value is fundamentally important in many areas of math, especially when solving equations as it helps us understand the distance or magnitude of values without considering their direction.
solving equations
Solving equations involves finding the value or values of variables that make the equation true. For absolute value equations, there are typically two scenarios to consider because the absolute value expression can represent two possible values (positive and negative). To solve the equation \(7|q| - 2 = 9\), you first isolate the absolute value expression, then deal with the positive and negative possibilities separately.
isolation of variables
Isolation of variables means manipulating an equation to get a specific variable by itself on one side of the equation. This simplifies the process of solving the equation. In the equation \(7|q| - 2 = 9\), we first add 2 to each side to isolate the absolute value term, giving \(7|q| = 11\). Then we divide each side by 7 to solve for \(|q|\), resulting in \(|q| = \frac{11}{7}\). This isolation helps to further simplify and solve the equation.
algebraic solutions
Algebraic solutions involve using algebraic methods and operations to solve equations. These methods include addition, subtraction, multiplication, and division. Once the absolute value is isolated and simplified, the equation \(|q| = \frac{11}{7}\) leads to two possible solutions. Because the absolute value represents both positive and negative values, we set up two separate equations: \(q = \frac{11}{7}\) and \(q = -\frac{11}{7}\). Solving these simple linear equations gives us the final solutions.

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