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Magazine Chapter 1: Problem 15
Solve each equation. $$ |2 x+5|=14 $$
Short Answer
Expert verified
x = 4.5 or x = -9.5.
Step by step solution
01
Identify the Absolute Value Equation
The given equation is \[ |2x + 5| = 14 \]. An absolute value equation can be split into two separate equations.
02
Separate into Two Equations
The absolute value equation \[ |2x + 5| = 14 \] is equivalent to the two linear equations: \[ 2x + 5 = 14 \] and \[ 2x + 5 = -14 \].
03
Solve the First Equation
Solve \[ 2x + 5 = 14 \] for \[ x \]:1. Subtract 5 from both sides: \[ 2x = 9 \].2. Divide both sides by 2: \[ x = \frac{9}{2} = 4.5 \].
04
Solve the Second Equation
Solve \[ 2x + 5 = -14 \] for \[ x \]:1. Subtract 5 from both sides: \[ 2x = -19 \].2. Divide both sides by 2: \[ x = \frac{-19}{2} = -9.5 \].
05
Write the Solution Set
The solutions to the equation \[ |2x + 5| = 14 \] are \[ x = 4.5 \] and \[ x = -9.5 \].
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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 is the distance of that number from zero on the number line, regardless of direction. It's always a positive value or zero. We denote absolute value using vertical bars, like this: |a| where a is any number.
For example:
|3| = 3
|鈭�3| = 3
In any absolute value equation like |2x + 5| = 14, both the expression inside the bars, 2x + 5, and its negative counterpart, -(2x + 5), are considered for solving because they are both 14 units away from zero.
For example:
|3| = 3
|鈭�3| = 3
In any absolute value equation like |2x + 5| = 14, both the expression inside the bars, 2x + 5, and its negative counterpart, -(2x + 5), are considered for solving because they are both 14 units away from zero.
linear equations
A linear equation is an equation that makes a straight line when it's graphed. The general form of a linear equation in one variable is ax + b = c. Here, a, b, and c are constants and x is the variable.
In our example, 2x + 5 = 14 is a linear equation. To solve such equations, we isolate the variable x.
Let's look at the process:
In our example, 2x + 5 = 14 is a linear equation. To solve such equations, we isolate the variable x.
Let's look at the process:
- First, subtract 5 from both sides: 2x = 14 - 5 which simplifies to 2x = 9.
- Second, divide both sides by 2: x = 9/2 = 4.5.
solution set
The solution set of an equation consists of all values that satisfy the equation. For absolute value equations like |2x + 5| = 14, we have to consider both the positive and negative cases. This means splitting the absolute value equation into two separate linear equations:
- 2x + 5 = 14
- 2x + 5 = -14
- 2x + 5 = 14 gives x = 4.5 .
- 2x + 5 = -14 gives x = -9.5 .
step-by-step solving
Solving absolute value equations can seem tricky, but breaking it down into steps makes it easier. Here鈥檚 a structured way to approach solving them:
- Step 1: Identify the Absolute Value Equation: Recognize the absolute value and the expression inside the bars.
- Step 2: Separate into Two Equations: Convert the absolute value equation into two linear equations to account for the positive and negative scenarios.
- Step 3: Solve the First Equation: Isolate the variable in the first linear equation through standard algebraic operations.
- Step 4: Solve the Second Equation: Similarly, solve the second linear equation by isolating the variable.
- Step 5: Write the Solution Set: Combine the solutions from both equations to form the complete solution set.
