Problem 49 - 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. $$ 5(x-2)-3(x+4) \geq 2 x-20 $$

Step-by-Step Solution

Answer

The solution set of the given inequality is an empty set, denoted by \(\varnothing\). This means that there are no values of \(x\) that can satisfy the given inequality.

Step 1: Distribute the numbers on both sides

The first step involves expanding the equation by multiplying the numbers with the variables and constants within the parentheses. After finishing this step, the inequality becomes: \(5x - 10 - 3x -12 \geq 2x - 20\)

Step 2: Simplify both sides

Next, we simplify both sides of the inequality by combining like terms. This will result in: \(2x - 22 \geq 2x - 20\)

Step 3: Isolate the variable

To solve for \(x\), subtract \(2x\) from both sides of the inequality, which gives: \(-22 \geq -20\) .

Step 4: Evaluate the inequality

To evaluate the inequality \(-22 \geq -20\) notice that this is a false statement since -22 is less than -20.

Step 5: Interpret the result

Because the inequality \(-22 \geq -20\) is false, there are no values of \(x\) that can satisfy it. The solution set is an empty set \(\varnothing\). We can graph this on a number line by not marking any points as no value of \(x\) will satisfy the inequality.

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. It provides a concise way to describe which numbers are included or excluded from a particular set. For example, the interval \(\left[a, b\right]\) includes all numbers between \(a\) and \(b\), including \(a\) and \(b\) themselves.
Here are some common symbols:

• \([a, b]\): All numbers between \(a\) and \(b\) (inclusive).
• \((a, b)\): All numbers between \(a\) and \(b\) (exclusive).
• \((a, b]\): Numbers greater than \(a\) but up to and including \(b\).
• \([a, b)\): Numbers including \(a\) but less than \(b\).

In this exercise, the solution set ended up being \(\varnothing\), which means there are no solutions. This is often represented by an empty interval.

Solution Sets

A solution set consists of all possible values that satisfy an equation or inequality. When solving linear inequalities, finding the solution set means determining all the values that make the inequality true.
In our exercise, after simplifying the inequality, we found that no values make it true, resulting in an empty set, represented by \(\varnothing\). This indicates there’s no solution.
It's important to carefully simplify the inequality to correctly identify the solution set. If the inequality ended up being true for any \(x\), those values would be part of the solution set.

Distributive Property

The distributive property is a fundamental algebraic principle that simplifies expressions involving parentheses. It states that \(a(b + c) = ab + ac\). This means you can multiply each term inside the parentheses by the factor outside.
In our exercise, we used the distributive property to expand \(5(x-2)\) into \(5x - 10\) and \(-3(x+4)\) into \(-3x - 12\).
By applying the distributive property, we made it easier to combine like terms and eventually simplify the inequality. Understanding this property helps in solving a wide range of algebraic problems.

Number Line

A number line is a visual representation of numbers in a straight line, helping to illustrate concepts like order and inequality. Each point on a number line corresponds to a number, and this helps in visualizing solution sets.
For inequalities, the number line is especially useful to show which numbers are included in the solution set.

• Open circles indicate numbers that are not included.
• Closed circles show numbers that are included.

In this scenario, since the solution was \(\varnothing\), the number line would have no shaded or marked points, reinforcing the fact that no solution exists. Using a number line can make complex concepts easier to understand and interpret.