Problem 10 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$|4 x+2|=5$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3}{4} \) and \( x = -\frac{7}{4} \).

Step 1: Set up the two cases for the absolute value equation

An absolute value equation \[|A| = B\] results in two separate equations: \[A = B\] and \[A = -B\]. Thus, for \[|4x + 2| = 5\], we have: \[4x + 2 = 5\] and \[4x + 2 = -5\].

Step 2: Solve the first equation

For \[4x + 2 = 5\], solve for \(x\): \[4x + 2 - 2 = 5 - 2\] \[4x = 3\] \[x = \frac{3}{4}\]

Step 3: Solve the second equation

For \[4x + 2 = -5\], solve for \(x\): \[4x + 2 - 2 = -5 - 2\] \[4x = -7\] \[x = \frac{-7}{4}\]

Step 4: Verify solutions

Substitute \( x = \frac{3}{4} \) and \( x = -\frac{7}{4} \) back into the original equation \[|4x + 2| = 5 \] to verify that both solutions satisfy the equation.

Key Concepts

solving absolute value equations

When solving absolute value equations, the goal is to isolate the absolute value expression and then set up two separate linear equations. Remember that an absolute value equation \[|A| = B\] leads to two possible equations: \[A = B\] and \[A = -B\]. Let's break down the steps using the given equation \[|4x + 2| = 5\]. To solve it, follow these steps:
Step 1: Remove the absolute value bars and consider both cases, so we get: \[4x + 2 = 5\] and \[4x + 2 = -5\].
Step 2: Solve each linear equation separately to find the possible values of \(x\).

• For \[4x + 2 = 5\]:
  Subtract 2 from both sides to get \[4x = 3\], then divide by 4, giving \[x = \frac{3}{4}\].
• For \[4x + 2 = -5\]:
  Subtract 2 from both sides to get \[4x = -7\], then divide by 4, giving \[x = -\frac{7}{4}\].

Step 3: Verify the solutions by substituting them back into the original equation to ensure both satisfy \[|4x + 2| = 5\].
This method will ensure you properly solve absolute value equations and verify the correctness of your solutions.

absolute value properties

Understanding the properties of absolute value is crucial in solving equations that involve them. Absolute value refers to the distance of a number from zero on a number line, ignoring any negative signs. Some key properties include:

• The absolute value of a positive number or zero is itself: \[|a| = a \,for\, a \geq 0\].
• The absolute value of a negative number is its positive counterpart: \[|a| = -a \,for\, a < 0\].
• Absolute values are always non-negative: \[|a| \geq 0\], meaning they can't be negative.
• If the absolute value of two expressions is equal, then the expressions inside can either be equal or negatives of each other.

To demonstrate, consider \[|4x + 2| = 5\]. This equation tells us that \[4x + 2\] is either 5 units away from zero or -5 units away. That's why we solve the two equations \[4x + 2 = 5\] and \[4x + 2 = -5\]. This property helps solve equations by handling each scenario separately.
Always remember to check all possible solutions in the original equation to ensure they are valid.

linear equations

Linear equations form the basis of solving absolute value equations. A linear equation is an equation of the first degree, meaning it has no exponents higher than one. It typically looks like \[ax + b = c\], where \[a\], \[b\], and \[c\] are constants.
To solve a linear equation, follow these steps:

• Isolate the variable term on one side of the equation by adding or subtracting constants from both sides.
• Next, divide by the coefficient of the variable to solve for it.
• Always perform the same operations on both sides to maintain equality.

Using the equation \[4x + 2 = 5\] as an example, we want to isolate \[x\]. First, subtract 2 from both sides to get \[4x = 3\], then divide by 4 to solve for \[x\], resulting in \[x = \frac{3}{4}\].
Similarly, for \[4x + 2 = -5\], subtract 2 from both sides to get \[4x = -7\], then divide by 4 to get \[x = -\frac{7}{4}\].
Linear equations are straightforward once you become familiar with these steps. They are the building blocks for more complex equations, like those involving absolute values.