Problem 46 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|4-4 x|+2=4$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

Step 1 - Isolate the absolute value expression

First, subtract 2 from both sides of the equation to isolate the absolute value expression: \[|4-4x| + 2 - 2 = 4 - 2\] This simplifies to: \[|4-4x| = 2\]

Step 2 - Set up two separate equations

Since \(|4-4x| = 2\), it means that the expression inside the absolute value can be 2 or -2. Therefore, set up the two equations: \[4-4x = 2\] and \[4-4x = -2\].

Step 3 - Solve for x in the first equation

Solve the equation \[4-4x = 2\]: 1. Subtract 4 from both sides: \[4 - 4x - 4 = 2 - 4\]2. Simplify to get: \[-4x = -2\]3. Divide by -4: \[x = \frac{-2}{-4} = \frac{1}{2}\]

Step 4 - Solve for x in the second equation

Solve the equation \[4-4x = -2\]: 1. Subtract 4 from both sides: \[4 - 4x - 4 = -2 - 4\]2. Simplify to get: \[-4x = -6\]3. Divide by -4: \[x = \frac{-6}{-4} = \frac{3}{2}\]

Step 5 - Combine solutions

The solutions for the equation \(|4-4x| + 2 = 4\) are \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

Key Concepts

absolute value

Absolute value measures the distance of a number from zero on the number line. It is always non-negative.
For example, the absolute value of both 3 and -3 is 3, written as \(|3| = 3\) and \(|-3| = 3\).
When we see an equation with an absolute value, it can represent two possible situations, because the distance can be achieved in two directions:

• The number itself
• The negative of the number

This is why absolute value equations often split into two separate equations during the solving process.

isolating expressions

Isolating an expression means you get the variable or the term you are interested in alone on one side of the equation.
This is often the first step in solving equations or inequalities.
In the given exercise, to isolate the absolute value expression:\(\text{|4 - 4x| + 2 = 4} \), you first subtract 2 from both sides of the equation.
This leaves you with:\(\text{|4 - 4x| = 2}\).
Isolating expressions helps simplify the problem and makes solving for the variable easier.

solving linear equations

Linear equations are equations of the first degree, which means they contain no exponents higher than one.
When solving linear equations, we aim to find the value of the variable.
From the given example, after isolating the absolute value, we split it into two linear equations:

• \(4 - 4x = 2\)
• \(4 - 4x = -2\)

For the first equation (\(4 - 4x = 2\)):

• Subtract 4 from both sides: \(4 - 4x - 4 = 2 - 4\)
• Simplify to: \(-4x = -2\)
• Divide both sides by -4: \(x = \frac{1}{2}\)

For the second equation (\(4 - 4x = -2\)):

• Subtract 4 from both sides: \(4 - 4x - 4 = -2 - 4\)
• Simplify to: \(-4x = -6\)
• Divide both sides by -4: \(x = \frac{3}{2}\)

This gives us the solutions \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\).

inequalities in algebra

Inequalities in algebra describe a relationship where one value is not necessarily equal to another.
They can show less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) relationships.
Absolute value inequalities follow a similar approach to equations, but we need to account for the range of values.
For instance, if you have an inequality involving an absolute value like \(|4 - 4x| < 2\), you break it into two inequalities:

• 4 - 4x < 2
• 4 - 4x > -2

Solving these gives you a range of values for x. However, our original exercise did not involve an inequality, but knowing how they work can be helpful for other problems.