Problem 43 - Precalculus: Functions and Graphs

Question

Solve the equation or inequality. Express the solutions in terms of intervals whenever possible. $$2|2 x+1|+1=19$$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -5\). Interval notation: \{-5, 4\}.

Step 1: Simplify the equation

First, isolate the absolute value expression on one side of the equation. Start by subtracting 1 from both sides of the equation: \[2|2x+1| + 1 - 1 = 19 - 1\]This simplifies to:\[2|2x+1| = 18\]

Step 2: Divide both sides by 2

Next, divide both sides of the equation by 2 to solve for the absolute value:\[\frac{2|2x+1|}{2} = \frac{18}{2}\]This results in:\[|2x+1| = 9\]

Step 3: Set up the equations for absolute value

The absolute value equation \(|2x+1| = 9\) means we need to consider two equations: one where the expression inside the absolute value is positive and one where it is negative. This gives us:1. \(2x+1 = 9\)2. \(2x+1 = -9\)

Step 4: Solve for x in the positive case

For the first equation, solve for \(x\):\[2x + 1 = 9\]Subtract 1 from both sides:\[2x = 8\]Now, divide both sides by 2:\[x = 4\]

Step 5: Solve for x in the negative case

For the second equation, solve for \(x\):\[2x + 1 = -9\]Subtract 1 from both sides:\[2x = -10\]Now, divide both sides by 2:\[x = -5\]

Step 6: Express the solution in interval notation

The solutions for \(x\) are \(x = 4\) and \(x = -5\). In interval notation, this does not create a true interval since these are discrete solutions. Thus, the solution set in interval format is expressed as:\[\{-5, 4\}\]

Key Concepts

Interval Notation

Interval notation is a way of describing sets of numbers, particularly intervals on the number line. It uses brackets and parentheses to indicate whether endpoints are included or not. In general,

• Use square brackets [ ] to include an endpoint in the interval, known as a closed interval.
• Use parentheses ( ) to exclude an endpoint, known as an open interval.
• Combine brackets and parentheses for half-open intervals.

In the case of solving the equation given, we ended up with discrete solutions: \(x = -5\) and \(x = 4\). These solutions do not form an interval, as intervals are defined by a continuous range of numbers. Therefore, these are expressed in set notation as \(\{-5, 4\}\), which simply lists the numbers separated by commas and enclosed in curly braces.

Solving Equations

Solving equations is about finding values for variables that make the equation true. The basic idea is to isolate the variable you are solving for. Here's a brief overview of the common steps:

• Start by simplifying the equation if necessary. This means removing parentheses or combining similar terms.
• Use operations like addition, subtraction, multiplication, or division to isolate the variable on one side of the equation.
• Check your solution by substituting the value back into the original equation.

In the exercise, we started by simplifying the left-hand side of the equation, then isolated the absolute value expression. The equation was then split into two separate equations to capture both the positive and negative scenarios of the absolute value. Solving each of these gave us the discrete solutions for \(x\).

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They can range from simple expressions like \(2x+1\) to more complex ones. In the context of solving equations, these are transformed and manipulated to isolate the variable:

• Identify the expression that needs to be simplified.
• Apply arithmetic operations to both sides equally to maintain the equality.
• Ensure the equation remains balanced by following the same operation on both sides.

In our example, the absolute value expression "\(2x+1\)" was crucial. It was initially masked by multiplication and addition but was simplified in stepwise fashion until isolated and broken into separate cases for resolution.

Discrete Solutions

Not all solutions form continuous intervals; some are discrete. Discrete solutions are individual values where the equation holds true. These can often be represented in set notation:

• Each solution is a unique point on the number line.
• They do not occur in a continuous sequence.
• Examples include counts of items, specific integer solutions, or points like \(\{-5, 4\}\).

For absolute value equations like the one we tackled, it is common to end up with discrete solutions. This is because solving \(|expression| = number\) splits the problem into two distinct paths, each potentially yielding a single unique solution. Hence, the final set, \(\{-5, 4\}\), shows these distinct points where the equation is satisfied.