Problem 86 - Algebra and Trigonometry

Question

Solve: $$ 6|1-2 x|-7=11 $$

Step-by-Step Solution

Answer

The solutions to the equation are x = -1 and x = 2.

Step 1: Add 7 to Both Sides

The first aim should be to isolate the absolute value term. To start with that, we should add 7 to both sides of the equation, which results in: 6|1-2x|= 18

Step 2: Divide Both Sides by 6

When dividing both sides by 6, we get the absolute value by itself, resulting in: |1-2x|= 3

Step 3: Write Two Separate Equations

Now, it is time to interpret the expression in absolute bars as positive and negative, yielding in two equations: 1-2x= 3 and 1-2x= -3

Step 4: Solve Each Equation

For each individual equation, 1 should be subtracted from both sides and the resulting expression should be divided by -2. Doing so will leave x by itself. Solving for x in both equations gives: x = -(3-1)/2 = -1 and x = -(-3-1)/2 = 2.

Key Concepts

Absolute Value

The concept of an absolute value relates to the distance a number is from zero on the number line, regardless of direction. In other words, the absolute value of a number is always non-negative. For example, the absolute value of both 5 and -5 is 5, symbolized in mathematics as \(|5| = 5\) and \(|-5| = 5\).

When we tackle absolute value in equations, we encounter a scenario where there are two possible cases. Since the absolute value of a number refers to its distance from zero without considering the direction, \(|x| = a\) actually represents two possible equations: \(x = a\) and \(x = -a\), assuming \(a\) is positive. This is because both a and -a would have the same absolute value. Understanding this concept is crucial when solving absolute value equations, as it leads to writing out two separate equations to solve, as is demonstrated in the original exercise.

Isolate Variable

The process of isolating the variable in an algebraic equation is fundamental to finding its solution. To 'isolate' means to get the variable on one side of the equation, so the other side consists of constants or numbers. This often involves performing a series of inverse operations to undo addition, subtraction, multiplication, or division.

For instance, if we have an equation like \(ax + b = c\), we first aim to undo the addition or subtraction. If \(b\) is added to \(x\), we subtract \(b\) from both sides, and vice versa if \(b\) is subtracted. Following this, to undo multiplication, we divide both sides by \(a\) to solve for \(x\). In this exercise, we first added 7 on both sides to start with isolating the absolute value term, followed by dividing by 6, which aptly demonstrates the technique of variable isolation.

Equation Solutions

Once the variable has been isolated, we need to solve for its value. For absolute value equations, this means setting up two separate but related equations that represent both possible values of the variable. Since absolute values can be thought of as distances, one equation will solve for the point one direction from zero, and the other equation will solve for the point in the opposite direction.

Using algebraic methods to solve these equations usually involves more inverse operations, such as subtracting constants from both sides or dividing by coefficients, until we reach the solutions to the variable. In our example, after isolating the absolute value, we then wrote two separate equations without the absolute value bars, taking into account the dual nature of absolute value, and solved them individually to find the two possible solutions for \(x\).

In summary, the solutions to an equation are the values of the variable that make the equation true, and for absolute value equations, there are typically two solutions to consider.