Problem 56 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -10 \leq 5 x<20 $$

Step-by-Step Solution

Answer

The solution set is \([-2, 4)\).

Step 1: Break Down the Compound Inequality

The compound inequality \(-10 \leq 5x < 20\) consists of two inequalities: \(-10 \leq 5x\) and \(5x < 20\). We will solve each part separately.

Step 2: Solve the First Inequality

To solve \(-10 \leq 5x\), divide both sides by 5:\[\frac{-10}{5} \leq x \-2 \leq x\]This simplifies to the inequality \(-2 \leq x\).

Step 3: Solve the Second Inequality

Next, solve \(5x < 20\). Again, divide both sides by 5:\[\frac{5x}{5} < \frac{20}{5} \x < 4\]This simplifies to \(x < 4\).

Step 4: Combine the Solutions

Combine the results from Step 2 and Step 3 to get the complete solution of the compound inequality:\[-2 \leq x < 4\]This represents all the values of \(x\) that satisfy both inequalities simultaneously.

Step 5: Represent the Solution in Interval Notation

In interval notation, the solution \(-2 \leq x < 4\) is represented as:\[[-2, 4)\]This means the interval includes \(-2\) but does not include \(4\).

Step 6: Graph the Solution Set

On a number line, graph the solution set by marking a solid dot on \(-2\) and an open dot on \(4\), with a line connecting them to indicate all numbers in between are included in the solution set.

Key Concepts

Interval Notation

Interval notation is a way to express the set of solutions for inequalities. It represents all numbers between specified limits without listing them explicitly. This method helps in clearly showing which endpoints are included or not within the solution set. When writing interval notation:

• Use a square bracket, "]" or "[", to indicate that a number is included in the set (also known as a closed interval).
• Use a parenthesis, ")" or "(", to show a number is not included (an open interval).

For instance, the solution to \( -2 \leq x < 4 \) is written as \([-2, 4)\).Here, \( [-2, 4) \) indicates that \(-2\) is part of the solution, while \(4\) is not. This not only simplifies the way we write solutions but also avoids any confusion about boundary inclusions.

Number Line Graph

A number line graph visually represents the solutions of inequalities on a horizontal line. It helps to quickly grasp which values are included or excluded in a solution set. To graph a set like \(-2 \leq x < 4\), follow these steps:

• Start at \(-2\) and place a solid dot, indicating that \(-2\) is included in the solution set.
• On \(4\), place an open dot to show that \(4\) is not included.
• Connect these points with a line, representing all the numbers between them as part of the solution.

This graphical representation gives a clear and straightforward view of the solution's range, making it easier to understand compound inequalities visually.

Solving Inequalities

Solving inequalities is about finding the range of values that a variable can take. Each part of the inequality must be solved separately, and the solutions are then combined. Here's a simple method:

• Break down a compound inequality like \(-10 \leq 5x < 20\) into two separate inequalities: \(-10 \leq 5x\) and \(5x < 20\).
• Solve each part individually. For example, divide each inequality by \(5\) to isolate \(x\). This gives \(-2 \leq x\) and \(x < 4\).
• Combine these solutions. The result tells us \(x\) must be greater than or equal to \(-2\) and less than \(4\).

By solving inequalities, we can determine all possible values for a variable that satisfy given conditions. This process is essential in algebra and helps to establish solution sets that can be represented in different notations.