Problem 44 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$7-\frac{4}{5} x<\frac{3}{5}$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 8\), which we represent in interval notation as \((8, +\inf)\), and we graph this on a number line by depicting all points greater than 8.

Step 1: Simplify the Inequality

Start by combining like terms on both sides, getting rid of fractions by multiplying the entire inequality by 5, which will not change the sign of inequality. This results in \(35 - 4x < 3\). Then, move \(4x\) to the right and \(3\) to the left by adding \(4x\) and subtracting \(3\) on both sides of inequality to get \(32 < 4x\). Divide by 4 to solve for \(x\), which gives us \(x > 8\).

Step 2: Express Solution in Interval Notation

The solution \(x > 8\) means that \(x\) can be any number greater than 8. But it cannot be exactly 8. So in interval notation, this is represented as \((8,+\inf)\), where the parentheses signify that the endpoint is not included in the interval.

Step 3: Graph the Solution on the Number Line

To draw a graph of the solution set on the number line, draw an open circle at 8 (since 8 is not included in the solution set) and draw an arrow to the right to signify that the solution includes all numbers greater than 8.

Step 4: Interpretation of Results

The solution set includes all real numbers greater than 8. So any number larger than 8 is a solution to the inequality \(7-\frac{4}{5}x<\frac{3}{5}\). The interval notation used to represent this solution helps to clearly indicate what values of \(x\) are part of the solution set, and the graph likewise provides a visual representation of the solution.