Problem 73 - College Algebra

Question

Solve absolute value inequality. \(|3 x-8|>7\)

Step-by-Step Solution

Answer

The solution for the absolute value inequality \(|3x-8| > 7\) is \(x > 5\) or \(x < 1/3\).

Step 1: Break down the absolute value inequality

Since absolute values can represent either positive or negative quantities, convert the absolute value inequality \(|3x-8| > 7\) into two separate inequalities: \(3x - 8 > 7\) and \(3x - 8 < -7\).

Step 2: Solve the first inequality

Start with the first equation \(3x - 8 > 7\). Add 8 to both sides of inequality, you will obtain \(3x > 15\). To isolate \(x\), divide both sides of the inequality by 3, \(x > 5\).

Step 3: Solve the second inequality

Then, solve the second equation \(3x - 8 < -7\). Similar to step 2, firstly add 8 to both sides, this results in \(3x < 1\). And when you then divide both sides by 3, you will obtain \(x < 1/3\).

Step 4: Combine Solutions

So the set of solutions for the given inequality are \(x > 5\) and \(x < 1/3\). These do still not overlap, so there are no solutions that will satisfy both conditions at the same time.

Key Concepts

Compound Inequalities

Compound inequalities involve taking two separate inequalities and solving them together. In the context of an absolute value inequality like \(|3x-8| > 7\), the compound inequalities come from separating the expression into two parts: one dealing with positive scenarios \((3x - 8 > 7)\), and the other handling the negative \((3x - 8 < -7)\). Each of these inequalities tells us something different about the possible values of \(x\).To solve compound inequalities:

• Split the absolute value inequality into two linear inequalities.
• Solve each inequality separately to get distinct solutions.
• Consider how the two solutions together form the complete solution set of the original problem.

Compound inequalities can indicate separate intervals or sometimes intersecting solutions. In this case, they define two distinct intervals on the number line. Understanding this concept is key for tackling these sorts of math problems and drawing accurate conclusions.

Inequality Solution

Solving inequalities involves finding the range of values that make the inequality true. This is like solving regular equations, but with an awareness that the solution represents a spectrum of values, rather than a single value.Here's how to solve linear inequalities step-by-step:

• Apply algebraic operations to isolate the variable on one side of the inequality.
• Do the same operation on both sides, maintaining the inequality's orientation.
• Remember that multiplying or dividing by a negative number flips the inequality sign.

The solutions obtained for the inequalities \(x > 5\) and \(x < \frac{1}{3}\) don't overlap in this case. Rather, they show two separate regions where each section of the inequality is satisfied. It's important to note that this means there is no value for \(x\) that satisfies both conditions simultaneously, leaving no solution when these are combined as they are part of an 'or' condition in the context of absolute inequalities.

Algebraic Expressions

Understanding algebraic expressions is crucial in solving equations and inequalities. An expression like \(3x-8\) involve terms (with coefficients and variables) combined by mathematical operations.Key steps to handle algebraic expressions:

• Identify each term and their coefficients (e.g., \(3x\) and \(-8\)).
• Use operations like addition or multiplication to simplify or rearrange the expression.
• Incorporate these expressions across different mathematical scenarios like equations or inequalities.

In solving \(|3x-8| > 7\), understanding how to manipulate \(3x-8\) is integral. This involves isolating \(x\) in each linear inequality to find the potential values \(x\) can take. Mastering algebraic expressions equips students with the ability to solve more complex mathematical problems effectively.