Problem 89 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality.. $$ 1<|2-3 x| $$

Step-by-Step Solution

Answer

The solution to the inequality \(1 < |2-3x|\) is \(x > 1\) or \(x < 1/3\).

Step 1: Understand the absolute value and split the inequality

Analyze the absolute value inequality \(1 < |2 - 3x|\). In this case, the absolute value can be split into two separate inequalities: \(2-3x > 1\) and \(2-3x < -1\). This translates to considering both situations where the value within the absolute value brackets is greater than 1 and also lesser than -1. This is due to absolute value always yielding a positive outcome, and thus, accounting for both positive and negative scenarios inside the absolute value sign.

Step 2: Solve the first inequality

Solve the first inequality \(2-3x > 1\). This is done by subtracting 2 from both sides of the equation, which yields \(-3x > -1\). To isolate x, divide both sides by -3. Important! When dividing by a negative number, the inequality sign must be flipped. This leads to \(x < 1/3\).

Step 3: Solve the second inequality

Solve the second inequality \(2-3x < -1\). Subtract 2 from both sides to get \(-3x < -3\). Then divide both sides by -3, remembering to flip the inequality sign, yielding \(x > 1\).

Step 4: Combine the solutions

The solution to the inequality is the union of the two solved inequalities. Write it as \(x > 1\) or \(x < 1/3\).

Key Concepts

Solving Inequalities

When solving inequalities, we aim to find all possible values of the variable that make the inequality true. Inequalities differ from equations in that they describe a range of potential solutions rather than a single outcome.

Here’s a simple way to understand solving inequalities:

• Perform the same operation on both sides of the inequality. This is similar to solving equations, where you would add, subtract, multiply, or divide both sides equally to isolate the variable.
• When you multiply or divide both sides of the inequality by a negative number, always reverse the direction of the inequality sign. This is because multiplying by a negative flips the order of numbers on a number line.

For instance, if solving the inequality \(-3x > -1\), after dividing both sides by \(-3\), make sure to switch the inequality sign to find \(x < \frac{1}{3}\). Pay extra attention to these steps, as they're vital for reaching the correct solution.

When you've solved each part of the inequality, consider the complete set of solutions. This often involves combining solutions from separate inequalities to describe when each results in a true statement.

Absolute Value

Absolute value represents the distance a number is from zero on the number line, regardless of direction, thus it’s always non-negative. The expression \(|a|\) is interpreted as follows:

• If \(a\) is positive or zero, \(|a| = a\).
• If \(a\) is negative, \(|a| = -a\).

To solve an absolute value inequality like \(1 < |2-3x|\),
remember:

• Split it into two separate inequalities: one considering the expression inside is more than the constant (positive scenario) and the other considering less than the negative of the constant (negative scenario).
• This helps to account for the absolute value's definition, ensuring both potential outcomes — whether the expression is positive or negative — are considered.

Working through these scenarios allows us to fully assess situations where the inequality holds for the absolute value. Through these steps, we get two cases to solve, namely \(2-3x > 1\) and \(2-3x < -1\).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the foundation for equations and inequalities, providing a compact way to express relations and problems.

In terms of solving absolute value inequalities, they play a crucial role. Take the expression \(2 - 3x\):

• This expression includes a constant term \(2\) and a variable term \(-3x\).
• Understanding how to manipulate these terms, like adding or subtracting constants and factoring or distributing variables, is essential for rearranging and solving equations or inequalities.

When dealing with inequalities such as the one in focus, it's crucial to consider each term separately to isolate the variable. For example, solving \(2-3x > 1\) involves subtracting \(2\) from both sides, then dividing by \(-3\), with a keen eye on flipping the inequality sign if dividing by a negative.

Through practice with algebraic expressions, students become adept at handling everything from basic calculations to more complex algebraic manipulations in equations and inequalities.