Problem 36 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 8 x+3>3(2 x+1)+x+5 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x>8\), or in interval notation, \( (8, \infty) \).

Step 1: Simplify the Equation

First, distribute 3 through \((2x+1)\) giving \(6x+3\). The inequality becomes \(8x+3>6x+3+x+5\). Combine like terms on both sides of the equation and get \(8x>7x+8\).

Step 2: Isolate the Variable

Subtract \(7x\) from both sides to isolate the variable \(x\) on one side of the inequality. This gives \(x>8\).

Step 3: Express the Solution Set in Interval Notation.

The solution \(x>8\) in interval notation is \( (8, \infty) \).

Step 4: Graph the Solution on a Number Line

On the number line, start at 8. Since x is greater than 8, make an open circle at 8 (since 8 is not included in the solution set) and draw an arrow to the right to indicate that x can be any value greater than 8.

Key Concepts

Interval Notation

Interval notation is a way of representing a range of numbers, typically used with inequalities. It provides a compact format to express solution sets. For the linear inequality given in our example, once simplified, the solution was determined to be \( x > 8 \).

In interval notation, this is expressed as \( (8, \infty) \). Here's how it works:

• Round brackets \((\) and \()\) indicate that the endpoint is not included in the set (known as an open interval).
• The comma separates the lower bound from the upper bound.
• \(\infty\) is used to express that there is no upper limit to the numbers in the interval.

Utilizing interval notation allows us to conveniently communicate the solution set of an inequality without an extended description.

Solution Sets

A solution set is a collection of all possible values that satisfy a given equation or inequality. For our inequality, the solution set includes all numbers greater than 8.

Let's delve into what makes up a solution set in general:

• The solution set should satisfy the original inequality. In this case, any number larger than 8 will fulfill \( 8x +3 > 3(2x+1) + x + 5 \).
• Solution sets can be finite (limited number of solutions) or infinite (unlimited number of solutions).
  
  Our solution set, being \((8, \infty)\), is infinite, meaning \(x\) can be any value greater than 8, continuing indefinitely.

Understanding solution sets is key to recognizing the range of values that adhere to an inequality.

Number Line

A number line is a visual representation of numbers in a linear format, where each point corresponds to a number. It's a helpful tool for illustrating solution sets of inequalities.

When graphing the solution \(x > 8\) on a number line:

• Begin at the point corresponding to 8.
• Use an open circle at 8 to indicate that 8 is not included in the solution set.
• Draw an arrow to the right to show that all numbers greater than 8 are included in the solution.

This graphical approach helps us immediately see which numbers are included or excluded, conveying the solution set in an intuitive format. Graphing on a number line complements the interval notation by providing a direct visual representation.