Problem 85 - Intermediate Algebra

Question

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. $$ 2 \geq 3|2-3 x|+2 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\). Graphically, it is a single point \(\frac{2}{3}\), represented as \(\left\{ \frac{2}{3} \right\}\) in interval notation.

Step 1: Isolate the Absolute Value

We start with the inequality \(2 \geq 3|2-3x| + 2\). To isolate \(|2-3x|\), subtract 2 from both sides of the inequality: \[2 - 2 \geq 3|2-3x|\] This simplifies to: \[0 \geq 3|2-3x|\]

Step 2: Divide by 3

Next, divide both sides by 3 to further isolate \(|2-3x|\): \[\frac{0}{3} \geq |2-3x|\] This simplifies to: \[0 \geq |2-3x|\]

Step 3: Solve the Absolute Inequality

The inequality \(0 \geq |2-3x|\) implies that the value of the absolute expression \(|2-3x|\) must be zero or negative. However, an absolute value expression \(|a|\) is always non-negative, meaning \(|a| \geq 0\). Hence, the only possibility here is: \[0 = |2-3x|\]

Step 4: Solve for x

Since \(|2-3x| = 0\), the expression inside the absolute value must be zero: \[2-3x = 0\] To solve for \(x\), subtract 2 from both sides: \[-3x = -2\] Then divide both sides by -3 to isolate \(x\): \[x = \frac{2}{3}\]

Step 5: Graph and Interval Notation

The solution to \(||2-3x| = 0\) is \(x = \frac{2}{3}\). In terms of graphing, this is a single point on the number line. Since there is only one solution, the interval notation for this solution is just \(\left\{ \frac{2}{3} \right\}\).

Key Concepts

Absolute Value Inequality

Understanding absolute value inequalities is key to solving many algebra problems. Absolute value refers to the distance of a number from zero on a number line, always resulting in a non-negative value. When you see an expression like \(|a|\), it denotes the absolute value of \`a\`. In our exercise, we encounter an absolute value inequality \(|2-3x|\).

To solve it, begin by isolating the absolute value on one side of the equation. This often involves first simplifying other parts of the inequality. It's important to remember the unique property of absolute values: they transform any expression inside them into its non-negative counterpart. That's why solving \(|2-3x| = 0\) results in just one solution, where the entire expression is zero.

This understanding leads directly to solving \(|2-3x| = 0\), demonstrating that you essentially treat the expression within the absolute value as equal to zero, because that's the only scenario that suits an inequality starting from zero.

Interval Notation

Interval notation is a way of writing subsets of the real number line and can help represent solutions to inequalities. In this particular problem, we arrive at a single value solution \(x = \frac{2}{3}\).

For single number solutions, interval notation distinctly uses curly braces to denote the point on the number line, like \(\left\{ \frac{2}{3} \right\}\). This points to the specific value \(x = \frac{2}{3}\) that satisfies the absolute value equality \(0 = |2-3x|\).

Another common use is when dealing with inequalities that have a range of solutions. In these cases, interval notation includes brackets and parentheses to indicate whether endpoints are included or excluded, respectively. Understanding how to transition from a solution to its interval notation representation is fundamental, especially when graphing solutions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. These expressions can become quite complex, especially when involving absolute values as seen in our exercise.

Here, the expression \(2-3x\) is encapsulated within an absolute value, which you need to address to solve for \(x\). The critical step in dealing with algebraic expressions is manipulating them to reveal simpler forms or solutions. In our case, solving starts with manipulating the initial inequality.

This includes distributing across terms, combining like terms, and isolating variables, all mainstays of solving algebra. Such skills are honed through practice and understanding how different algebraic manipulations affect the overall expression. The ultimate goal is to arrive at a clear, concise solution that unambiguously communicates the relationship between variables.