/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Solve. Let \(f(x)=|2 x+6| .\) ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 47

Solve. Let \(f(x)=|2 x+6| .\) Find all \(x\) for which \(f(x)=8\)

Short Answer

Expert verified
The solutions are \( x = 1 \) and \( x = -7 \).

Step by step solution

01

Set up the equation

The given function is \( f(x) = |2x + 6| \). To find the values of \( x \) such that \( f(x) = 8 \), set \( |2x + 6| = 8 \).
02

Remove the absolute value

Since the absolute value equation \( |a| = b \) can be split into two cases, we write:1) \( 2x + 6 = 8 \)2) \( 2x + 6 = -8 \)
03

Solve the first equation

Solve \( 2x + 6 = 8 \) by isolating \( x \): \Subtract 6 from both sides: \[ 2x = 2 \]Divide both sides by 2: \[ x = 1 \]
04

Solve the second equation

Solve \( 2x + 6 = -8 \) similarly:Subtract 6 from both sides: \[ 2x = -14 \]Divide both sides by 2: \[ x = -7 \]
05

List all solutions

The solutions \( x = 1 \) and \( x = -7 \) satisfy the equation \( f(x) = 8 \).

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Key Concepts

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

Solving Absolute Value Equations
Absolute value equations may seem tricky, but they are easier than you think! An absolute value equation is an equation where the unknown variable is inside an absolute value. When we say absolute value, we mean the distance of a number from zero, no matter which direction you go on the number line.
To solve these equations, we need to set up two separate cases. Absolute value means that whatever is inside the absolute value symbols can be equal to the number or its opposite. For example, if the equation is \(|x| = 3\), then
  • \ x = 3
  • x = -3
So, always remember: every absolute value equation splits into two simpler equations to solve!.
Algebraic Equations
When you encounter an algebraic equation, you're working with a mathematical statement that shows the equality between two expressions. In our case, the expression is inside the absolute value symbols. The goal is simple: find the value of the variable that makes the equation true!
To solve an algebraic equation:
  • Isolate the variable.
  • Simplify both sides of the equation.
For example, take our absolute value equation from the problem: \(|2x + 6| = 8\).
We split it into two straightforward algebraic equations:
  • 2x + 6 = 8
  • 2x + 6 = -8
From here, we can solve them by isolating x.
Easy, right?.
Finding Solutions
Finding solutions to absolute value equations is all about breaking it down into manageable pieces. Let's take another look at our equations from earlier:
  • 2x + 6 = 8
  • 2x + 6 = -8
For both cases, we want to solve for x. First, subtract 6 from both sides:
  • 2x = 2 \implies x = 1
  • 2x = -14 \implies x = -7
That gives us our solutions: \(x = 1\) and \(x = -7\).
These are the points where the absolute value function reaches 8!
Make sure to always check your solutions in the original equation to confirm they're correct!.

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