Problem 33 - 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. $$ 8 x-11 \leq 3 x-13 $$

Step-by-Step Solution

Answer

The solution set for the inequality is \(x \leq -0.4\)

Step 1: Simplify the Inequality

We commence by transferring all terms including \(x\) to one side of the inequality and all the constant terms to the other side. We subtract \(3x\) from both sides: \(8x - 3x \leq -13 +11\)

Step 2: Simplify Both Sides

We then simplify both sides of the inequality to get \(5x \leq -2\)

Step 3: Final Solution

Finally, to isolate \(x\), we divide both sides of the inequality by 5 to get \(x \leq -0.4\)

Step 4: Graph the Solution on a Number Line

In the number line, place a solid circle over -0.4 to signify inclusivity, and fill in the line up to -0.4 in the negative direction, indicating that all numbers less than or equal to -0.4 are valid solutions

Key Concepts

Interval Notation

Interval notation provides a concise way of writing sets of numbers that comprise all the numbers between a starting point and an endpoint. It is commonly used to express the solution sets of inequalities. In interval notation, brackets and parentheses have specific meanings. Brackets, such as \[ or \], are used to indicate that the endpoint is included in the set, known as a 'closed interval'. Parentheses, such as \( or \) are used for 'open intervals', where the endpoint is not included.

For example, the solution of the inequality \( x \leq -0.4 \) would be written in interval notation as \( (-\infty, -0.4] \). Here the square bracket indicates that \(-0.4\) is part of the solution set, while the parenthesis next to \(-\infty\) indicates that the set extends indefinitely in the negative direction without including infinity, as infinity is not a real number.

Number Line Graphing

Graphing on a number line is a visual method to represent the set of all possible solutions to an inequality. When graphing the solution set of a linear inequality, we usually use a filled-in circle or dot to represent that a number is included in the set (closed interval), or an open circle to show that a number is not part of the solution set (open interval).

An inequality such as \( x \leq -0.4 \) is represented on a number line with a filled-in circle at \( -0.4 \) and a shaded line extending to the left to signify all numbers smaller than \(-0.4\) are included. It’s a straightforward way to see at a glance what the solution set includes.

Inequality Simplification

Simplifying an inequality is a process similar to solving equations. The goal is to isolate the variable on one side to make the inequality easier to understand and solve. This often involves combining like terms and performing inverse operations.

In the given exercise, the terms containing \(x\) were moved to one side and the constants to the other, resulting in \(5x \leq -2\). The next step is to isolate \(x\) by dividing both sides by the coefficient of \(x\), which, in this case, is 5. This series of steps transforms the original inequality into its simplest form, allowing us to easily find the solution set or graph it on a number line.

Linear Inequality Solutions

The solutions to linear inequalities are found by manipulating the inequality into a form where the variable is isolated on one side. Unlike equations, inequalities can have a range of solutions that are represented as intervals. After finding the inequality’s simplest form, as we did with \(5x \leq -2\) which simplified to \(x \leq -0.4\), we interpret this to mean that any real number less than or equal to \(-0.4\) will satisfy the inequality.

Linear inequalities must be treated carefully when multiplied or divided by negative numbers, as this action inverses the inequality symbol (for example, changing a \(<\) to a \(>\)). When solved correctly, the solution set can be expressed in interval notation, graphically represented on a number line, or described in words, providing a comprehensive view of all possible solutions.