Problem 70 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(|4 x+3|=9\)

Step-by-Step Solution

Answer

The solutions to the equation \(|4x+3| = 9\) are \(x = 1.5\) and \(x = -3\).

Step 1: Establish Two Potential Equations

Since the result of the operation inside the absolute value can either be positive or negative, the equation can be written as two potential equations: \(4x + 3 = 9\) and \(4x + 3 = -9\).

Step 2: Solve First equation \(4x + 3 = 9\)

Subtract 3 from both sides of the equation to isolate \(4x\), resulting in a new equation: \(4x = 9 - 3\). Solve this for \(x\) by dividing both sides by 4. The solution is \(x = 6/4\), or \(x = 1.5\).

Step 3: Solve Second Equation \(4x + 3 = -9\)

Just like in Step 2, subtract 3 from both sides to isolate \(4x\), resulting in the equation \(4x = -9 - 3\). Solve this for \(x\) by dividing both sides by 4. The solution is \(x = -12/4\), or \(x = -3\).

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics that denotes how far a number is from zero on the number line, regardless of direction. It transforms any number into a positive value. This means if you have an equation like \(|x| = a\), where \(a\) is positive, \(x\) can be either \(a\) or \(-a\). For example, in the equation \(|4x + 3| = 9\), the expression inside the absolute value, \(4x + 3\), could either be 9 or -9. This is because both values have the same distance from zero, emphasizing absolute value's role in considering both positive and negative directions.

Breaking Down Linear Equations

Linear equations are algebraic expressions that make straight lines when graphed on a coordinate plane. They typically take the form \(Ax + B = C\), where \(A\), \(B\), and \(C\) are constants, and \(x\) represents the variable to be solved. In the context of solving an absolute value equation, linear equations emerge when we consider each possible outcome separately. For instance, in the given exercise, the outcomes \(4x + 3 = 9\) and \(4x + 3 = -9\) are linear equations. These set the groundwork for finding specific values of \(x\), preserving the linear nature of the expressions on either side.

Following a Step-by-Step Solution

Solving equations step-by-step ensures understanding and accuracy. It involves simplifying equations and isolating the variable step by step. In this exercise, start by setting up two separate linear equations: \(4x + 3 = 9\) and \(4x + 3 = -9\).

• Begin with the first equation, \(4x + 3 = 9\). Subtract 3 from both sides to get \(4x = 6\). Finally, divide by 4, resulting in \(x = 1.5\).
• Next, tackle \(4x + 3 = -9\) by similarly subtracting 3 from both sides, simplifying to \(4x = -12\). Divide by 4 to find \(x = -3\).

This logical progression helps in systematically approaching solutions and teaches important skills in algebraic manipulation. By practicing such detailed solutions, one enhances their problem-solving abilities, crucial for mathematics and beyond.