Question

Solve the equation. $$|4 x-2|=22$$

Short answer

The solutions to the equation are \(x = 6\) and \(x = -5\).

Step-by-step solution

Split the Equation

Split the equation into two parts, according to the properties of absolute value: \(4x-2 = 22\) and \(4x-2 = -22\).

Solve the First Equation

Solve \(4x-2 = 22\) for \(x\). Start by adding \(2\) to both sides to get \(4x = 24\). Then divide both sides by \(4\) to find \(x = 6\)

Solve the Second Equation

Solve \(4x-2 = -22\) for \(x\). Add \(2\) to both sides to get \(4x = -20\). Divide by \(4\) to find \(x = -5\)

Key concepts

Absolute Value Properties

Understanding the properties of absolute value is essential when solving equations that include absolute value expressions. An absolute value measures the distance of a number from zero on the number line, regardless of direction, which is why it is always non-negative. The absolute value of a number, denoted as |a|, will be the number itself if the number is positive or zero, and the opposite of the number if it is negative.

One key property that helps in solving absolute value equations is that if |a| = b where b is a positive number, then a must be either b or -b. This is because both b and -b are at the same distance from zero. Hence, the absolute value equation |4x - 2| = 22 becomes two separate linear equations: 4x - 2 = 22 and 4x - 2 = -22. This dual nature of the solution is at the heart of solving these types of equations, and explains why we need to consider two different cases during the process.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. These equations form a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable, like x, can be written as ax + b = c, where a, b, and c are constants.

The goal when solving linear equations is to isolate the variable on one side of the equation, which allows us to find its value. For example, given the equation 4x - 2 = 22, we aim to get x by itself. We do this by performing operations that will 'undo' whatever is being done to x. Starting by adding 2 to both sides to 'cancel out' the minus 2, we get 4x = 24. Then, dividing both sides by 4 isolates the x, giving us x = 6. These simple steps are the building blocks of algebraic problem solving and apply to a plethora of different situations.

Algebraic Problem Solving

Algebraic problem solving involves a systematic approach to work through equations and find unknowns. It's not just about knowing the operations but understanding when and how to apply them effectively. When faced with an absolute value equation like |4x - 2| = 22, breaking down the problem into more manageable parts is a strategic move. This is why we split the original equation into two simpler linear equations based on the properties of absolute values.

Ensuring all steps are logically grounded and each operation is reversible helps maintain the equation's balance. Adding or subtracting the same number from both sides, like we do when we add 2 to control the '-2' in our equation, or multiplying/dividing by the same non-zero number, such as dividing by 4 to simplify '4x', are examples of these reversible operations. To check if the solutions are correct, they can be substituted back into the original equation. This not only verifies the accuracy of the solutions but also reinforces the understanding of algebraic foundations.