Problem 60 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|7+2 x|=0$$

Step-by-Step Solution

Answer

x = -\(\frac{7}{2}\)

Step 1: Understanding absolute value equations

Recall that the absolute value \(|x|\) of a number is always non-negative (i.e., \(|x| \geq 0\)). Therefore, for an absolute value equation \(|A| = 0\), the inside of the absolute value must be 0 because the only number whose absolute value is 0 is 0 itself.

Step 2: Set the inside equal to zero

Set the expression inside the absolute value, \(7 + 2x\), equal to zero: \(7 + 2x = 0\).

Step 3: Solve for x

To solve for \(x\), isolate \(x\) on one side of the equation. Subtract 7 from both sides to get: \2x = -7\. Now divide both sides by 2: \x = -7/2\.

Key Concepts

Solving Absolute Value Equations

Absolute value equations can be tricky, but with the right approach, they become straightforward.
The absolute value of a number refers to its distance from zero on the number line, which is always non-negative. Hence, for equations that involve absolute values, we need to understand the core principle that \(|x| \geq 0\).

To solve an absolute value equation like \(|7 + 2x| = 0\), remember:

• The expression inside the absolute value must be zero because the absolute value is zero only when the number itself is zero.
• So, we translate the equation to \(|7 + 2x| = 0\) to \(|7 + 2x = 0|\).

From there, we can solve for the variable by standard algebraic methods.

Isolating Variables

Isolating the variable in an equation simplifies solving it.
In basic algebra, isolating the variable means getting the variable by itself on one side of the equation.

Consider the equation \(|7 + 2x| = 0\), simplified as \(|7 + 2x = 0|\). Our goal is to isolate \(|x|\):

• First, eliminate constants on the side of the variable. Subtract 7 from both sides: \(|7 + 2x - 7 = -7|\), simplifying to \(|2x = -7|\).
• Next, simplify further by dividing by the coefficient of \(|x|\). Divide both sides by 2: \(|2x/2 = -7/2|\), giving \(|x = -7/2|\).

Respect these steps in order, and isolating variables will help in solving the equations smoothly.

Linear Equations

Linear equations are foundational in algebra.
These equations graph as straight lines and have the standard form \(|ax + b = c|\). They are solved by isolating the variable using inverse operations.

For example, in \(|7 + 2x| = 0\):

• First, recognize it is a linear equation where \(|a = 2|\), \(|b = 7|\), and \(|c = 0|\).
• Set the inside of the absolute value equal to zero, i.e., \(|7 + 2x| = 0|\), resulting in \(|7 + 2x = 0|\).
• Next, perform operations to isolate \(|x|\). Subtract 7: \(|7 - 7 + 2x = 0 - 7|\), so we get \(|2x = -7|\).
• Now, divide both sides by 2: \(|2x/2 = -7/2|\), yielding \(|x = -7/2|\).

Lorem ipsum dolor sit, this satisfies the structure of a linear equation solution.