Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, solve for \(x\). $$ |4 x+3|=7 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -\frac{5}{2}\).

Step 1 - Understand the Absolute Value Property

Recall that for any real number, the absolute value equation \(|a| = b\) implies \(a = b\) or \(a = -b\).

Step 2 - Set up Two Equations

Using the absolute value property, split \(|4x + 3| = 7\) into two separate equations: 1. \(4x + 3 = 7\)2. \(4x + 3 = -7\)

Step 3 - Solve the First Equation

First, solve \(4x + 3 = 7\):Subtract 3 from both sides:\[4x = 7 - 3\]\[4x = 4\]Divide by 4:\[x = 1\]

Step 4 - Solve the Second Equation

Next, solve \(4x + 3 = -7\):Subtract 3 from both sides:\[4x = -7 - 3\]\[4x = -10\]Divide by 4:\[x = -\frac{10}{4}\]\[x = -\frac{5}{2}\]

Step 5 - State the Solutions

The solutions to the equation \(|4x + 3| = 7\) are \(x = 1\) and \(x = -\frac{5}{2}\).

Key Concepts

absolute value property

The absolute value property is a fundamental concept in algebra. It states that the absolute value of a number is always non-negative and represents its distance from zero on a number line. For any real number, \(|a| = b\) means that\( a = b \) or \( a = -b \). Remember, the absolute value of a number represents its distance from zero; therefore, whether a number is positive or negative, its absolute value is the same. For example, \(|3| = 3\) and \(|-3| = 3\). Applying this property to absolute value equations helps to split the problem into two separate linear equations.

solving linear equations

Solving linear equations is crucial when dealing with absolute value problems. Linear equations form the basics of most algebraic operations. Let's break this down using our example: \(|4x + 3| = 7\). First, understand that we need to split the absolute value equation into two linear equations. After splitting, we get: \(4x + 3 = 7\) and \(4x + 3 = -7\). Solving these linear equations involves basic algebraic steps: isolate the variable on one side of the equation by performing inverse operations. For instance, to solve \(4x + 3 = 7\), subtract 3 from both sides to get \(4x = 4\). Then, divide both sides by 4 to find \(x = 1\). Practice these steps to get better at breaking down and solving linear equations.

splitting absolute value equations

Splitting absolute value equations is a method used to simplify and solve absolute value problems. When you have an equation like \(|4x + 3| = 7\), you essentially need to consider every possible scenario that would make the absolute value expression true. Therefore, this results in two different linear equations to solve: \(4x + 3 = 7\) and \(4x + 3 = -7\). Just follow these steps:

• First, split the absolute value equation into two parts.
• Second, solve each resulting linear equation separately.
• Lastly, combine the solutions for the final answer.

This method helps to ensure that all possible solutions for x are covered. In our example, solving the second part, \(4x + 3 = -7\), involves similar steps: subtract 3 from both sides to get \(4x = -10\), then divide by 4 resulting in \(x = -\frac{5}{2}\). The final solutions are \(x = 1\) and \(x = -\frac{5}{2}\).