Problem 48 - 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. $$ 3(x-8)-2(10-x)>5(x-1) $$

Step-by-Step Solution

Answer

\(\varnothing\)

Step 1: Distribute

First, distribute across parentheses on both sides to simplify the equation. So, \(3(x-8)-2(10-x)\) would become \(3x - 24 - 20 + 2x\) and \(5(x-1)\) would become \(5x - 5\).

Step 2: Simplify

Next, combine like terms on both sides of the inequality. On the left side, combine \(3x\) and \(2x\) together and \(-24\) and \(-20\) together to get \(5x - 44\). On the right side, the equation remains \(5x - 5\). The inequality becomes \(5x - 44 > 5x - 5\).

Step 3: Subtract 5x from both sides

In the next step, subtract \(5x\) from both sides of the inequality to keep the equation balanced. This results in the simplified inequality \(-44 > -5\).

Step 4: Interpretation

The inequality \(-44 > -5\) is incorrect, which means there is no solution for the inequality \(3(x-8)-2(10-x)>5(x-1)\).

Step 5: Graph and Interval Notation

Since there is no solution, the graph on the number line would contain no points. In interval notation, the solution set is represented by \(\varnothing\), the symbol for an empty set.

Key Concepts

Interval Notation

When representing solutions to inequalities, interval notation provides a clear and concise way to express the set of numbers that are included in the solution. This system uses brackets and parentheses to describe intervals on the number line. In interval notation, a square bracket \[ or \] indicates that the number is included in the set, known as a 'closed' interval. On the other hand, a parenthesis \(( or \)) means the number is not included, referred to as an 'open' interval. For example, the interval \([1, 5)\) includes 1 but does not include 5. Special cases also arise, such as an empty set, denoted by \(\varnothing\), which signifies no real numbers satisfy the inequality, as is the case in our example inequality.
Interval notation is not only compact, but it also simplifies the process of communicating complex sets of numbers, which can be particularly useful in situations involving unions or intersections of sets.

Graphing Solution Sets

Graphing solution sets on a number line is an integral part of understanding linear inequalities. After solving an inequality, representing the solution graphically can provide a visual comprehension of the range of values that are solutions. To perform this graphing, you place a solid circle on the number line for numbers that are 'included' in the solution and an open circle for 'excluded' numbers. Then, a line is drawn between these to illustrate the continuum of numbers that make the inequality true. In our example wherein no solution exists, you would simply see a number line with no marked points, indicating that there is no set of numbers that will satisfy the inequality.

Linear Inequality Algebra

Linear inequality algebra involves techniques similar to those used in linear equations, but with special consideration to the inequality sign. Solving a linear inequality generally involves distributing terms, combining like terms, and isolating the variable on one side. However, one crucial difference from equations is the need to reverse the inequality sign when multiplying or dividing both sides by a negative number. In the case of the given example \(3(x-8)-2(10-x)>5(x-1)\), we simplify and find that the inequality simplifies to an incorrect statement \(-44 > -5\), indicating that there are no possible solutions. No need to reverse the sign since we never multiplied or divided by a negative number in our steps.

Number Line Representations

The number line is a valuable tool for visualizing and representing the solutions to inequalities. Number line representations can clarify where the solutions lie in relation to real numbers. A number line is marked with points or intervals that indicate where the variable x satisfies the inequality. Solid dots indicate that a number is a part of the solution set, while open dots mean the number is not included. In instances where no solution is available, as with our example, you would normally label the number line, but you would not highlight or mark any intervals or points since there are no values of x that solve the inequality.