Problem 1 - Algebra and Trigonometry

Question

Solve the equation. $$ |4 x|=24 $$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -6\).

Step 1: Understand the Problem

We need to solve the equation \(|4x|=24\), which involves an absolute value expression. An absolute value\(|a|\) gives the distance of \(a\) from 0 on a number line, so it's always non-negative.

Step 2: Set up Two Equations

When dealing with absolute values, we consider two cases: the expression inside the absolute value is either positive or negative. Thus, we set up two equations: 1. \(4x = 24\)2. \(4x = -24\)

Step 3: Solve the First Equation

Solve the equation \(4x = 24\):- Divide both sides by 4 to isolate \(x\).- \(x = \frac{24}{4}\)- \(x = 6\)

Step 4: Solve the Second Equation

Solve the equation \(4x = -24\):- Divide both sides by 4 to isolate \(x\).- \(x = \frac{-24}{4}\)- \(x = -6\)

Step 5: Check the Solutions

Substitute \(x = 6\) into the original equation: - \(|4(6)| = 24\)- Since \(|24| = 24\), \(x = 6\) is a valid solution. Substitute \(x = -6\) into the original equation:- \(|4(-6)| = 24\)- Since \(|-24| = 24\), \(x = -6\) is also a valid solution.

Step 6: Write the Final Solution Set

The solutions to \(|4x| = 24\) are \(x = 6\) and \(x = -6\). Hence, the solution set is \(\{6, -6\}\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra. It involves finding the value or set of values for variables that make an equation true. In our example, the equation we are solving is \(|4x| = 24\). Here, the goal is to find all possible values of \(x\) that will satisfy this equation.
When solving equations, it is crucial to isolate the variable. In simple terms, this means getting the variable alone on one side of the equation. For the equation \(4x = 24\), isolating \(x\) involves dividing both sides by 4 to find \(x = 6\). We perform similar steps for the equation \(4x = -24\), giving us \(x = -6\).

• Isolate the variable by performing arithmetic operations.
• Check your solutions by substituting back into the original equation.

This verification ensures that both solutions, 6 and -6, satisfy the original equation.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition and multiplication). In the expression \(4x\), we have the variable \(x\) being multiplied by 4. It's crucial to understand expressions as these are the building blocks of equations.
In algebra, you will encounter many types of expressions. Here, understanding \(4x\) means recognizing that any number substituted for \(x\) will be multiplied by 4. This concept is helpful when you set up your equations to solve for the values of \(x\). For instance, to solve \(4x = 24\), it's intuitive to reverse the operation (multiplication) by dividing both sides by 4.

• Identify the operation applied to the variable (here, multiplication by 4).
• Use inverse operations to solve equations (division in this case).

Grasping these fundamental concepts of algebraic expressions makes solving equations much more manageable.

Equations with Absolute Values

Equations with absolute values involve expressions that measure the distance of a number from zero. The absolute value, denoted by \(|a|\), signifies that it is always non-negative, regardless of whether \(a\) is positive or negative.
When you solve absolute value equations, like \(|4x| = 24\), you must consider that \(4x\) can equal 24 or -24 because both, when inside the absolute value, result in 24. So, you solve two separate equations, first setting \(4x = 24\) and then \(4x = -24\).

• Recognize that absolute values imply two potential equations: one for the positive and one for the negative scenario.
• Solve each scenario independently to find all possible solutions.
• Verify each solution by substituting back into the original equation to ensure correctness.

By understanding these properties of absolute values, you can effectively handle and solve absolute value equations.