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Chapter 1: Problem 103

Solve each equation or inequality. $$ |4 x+1|=0 $$

Short Answer

Expert verified
x = -\frac{1}{4}.

Step by step solution

01

Understand the Absolute Value Property

The absolute value of a number is its distance from zero on the number line. For any real number, if \(|a| = 0\), then \[a = 0.\]
02

Apply the Absolute Value Property

Given \[|4x + 1| = 0\], use the property from Step 1 to set the expression inside the absolute value to 0: \[4x + 1 = 0.\]
03

Solve for x

Now solve the equation \[4x + 1 = 0\]:\[4x + 1 = 0\]Subtract 1 from both sides: \[4x = -1\]Divide both sides by 4: \[x = -\frac{1}{4}.\]

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Key Concepts

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

absolute value
The concept of absolute value centers on the distance of a number from zero on the number line. It's always zero or a positive number. Here are some important points about absolute value:
鈥� \[ |a| = a \] if \[ a \geq 0 \] (if the number is positive or zero)
鈥� \[ |a| = -a \] if \[ a < 0 \] (if the number is negative, absolute value makes it positive)
In our exercise, we need to solve \[ |4x + 1| = 0 \]
Since absolute value equals zero only when the number itself is zero, we have to set the inside of the absolute value expression to zero: \[ 4x + 1 = 0 \]
linear equations
Linear equations are equations of the first degree, which means the variable's highest power is one. They form straight lines when graphed. The general form is \[ ax + b = 0 \], where \[ a \] and \[ b \] are constants.
In our case, once we apply the absolute value property, we get the linear equation \[ 4x + 1 = 0 \]
To solve this, follow these steps:
1. Subtract 1 from both sides: \[ 4x + 1 - 1 = 0 - 1 \]
Simplifies to: \[ 4x = -1 \]
2. Divide both sides by 4 to isolate \[ x \]:
\[ x = -\frac{1}{4} \]
algebraic solutions
An algebraic solution involves manipulating the equation using algebraic rules to find the value(s) of the variable. Steps to achieve an algebraic solution include:
鈥� Isolating the variable
鈥� Applying inverse operations (adding, subtracting, multiplying, dividing)
鈥� Simplifying the equation
In our exercise:
1. After isolating \[4x + 1 \], subtract 1:
\[ 4x = -1 \]
2. Divide by 4:
\[ x = - \frac{1}{4} \]
This gives us the solution: \[ x = - \frac{1}{4} \]. By following these steps, we find the value of \[ x \].
step-by-step algebra
Step-by-step algebra involves solving equations methodically. Breaking down complex problems into smaller, manageable steps makes it easier to understand and solve them. Always:
1. Understand the problem
2. Apply the relevant properties and rules
3. Simplify as needed
Let's apply this process to our problem:
1. Recognize the absolute value equation \[ |4x + 1| = 0 \] and know it implies \[ 4x + 1 = 0 \]
2. Isolate the variable \[ x \] by subtracting 1 and dividing by 4:
\[ 4x + 1 = 0 \]
\[ 4x = -1 \]
\[ x = -\frac{1}{4} \]
By following these straightforward steps, algebra becomes easier to tackle.

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Most popular questions from this chapter

Solve each equation or inequality. $$ |4 x-1| \leq 0 $$

Give, in interval notation, the unknown numbers in each description. One third of a number is added to 6 , giving a result of at least 3 .

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Dr. Mosely has determined that \(99 \%\) of the babies he has delivered have weighed \(x\) pounds, where $$ |x-8.3|<1.5 $$ What range of weights corresponds to this inequality?
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