Problem 37 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2 \leq x<5$$

Step-by-Step Solution

Answer

The graph represented has a filled circle at -2 and an empty circle at 5 with the line between -2 and 5 shaded, representing all the solutions to the given inequality.

Step 1: Understanding Inequalities

Analyze the given inequality: \( -2 \leq x<5 \). This means `x` can take any value between -2 and 5, including -2 but excluding 5. It's important to remember, the symbol \(\leq\) means 'less than or equal to', so the value -2 is included, but \(<\) symbol means 'less than' which excludes the value 5.

Step 2: Plotting on Number Line

Start plotting the inequality on a number line. Mark the point -2 and 5 on the number line. Since -2 is included in the solution set, represent this by a closed point or circle at -2. Since 5 is not included in the solution set, represent this by an open point or circle at 5.

Step 3: Highlighting the Solution Set

The values of `x` which satisfy the inequality are all those values lying between -2 and 5. So, to represent these values, shade or highlight the portion of the number line that lies between -2 (included) and 5 (excluded).

Key Concepts

Solution Set

A solution set is a collection of values that satisfy an equation or inequality. In our exercise, the solution set consists of all values of \(x\) that lie between -2 and 5. - The inequality \(-2 \leq x < 5\) indicates that \(x\) includes -2, as suggested by the "less than or equal to" symbol (\(\leq\)).- It does not include 5, as shown by the "less than" symbol (<).Therefore, the solution set is all the real numbers \(x\) such that \(-2 \leq x < 5\). This range contains every real number between -2 and 5, inclusive of -2 but not including 5. Always ensure to carefully interpret inequality symbols to correctly identify the solution set.

Number Line

A number line is a visual tool used to show the position of numbers relative to each other. It is extremely useful for graphing inequalities because it helps us easily identify the range of solutions. On a number line:- Every point corresponds to a number.- Values increase as you move from left to right.- Negative numbers appear on the left of zero, while positive numbers are on the right.To graph the solution set of the given inequality \(-2 \leq x < 5\), mark points at -2 and 5 on the number line. These points indicate the boundaries of the solution set. Using a number line effectively shows you which numbers are part of the solution set and which are not.

Inequality Symbols

Inequality symbols are crucial for interpreting mathematical inequalities. These symbols indicate the relationship between different values. - The symbol \(\leq\) means "less than or equal to". For example, \(-2 \leq x\) includes -2 as a possible value of \(x\).- The symbol < denotes "less than". Therefore, \(x<5\) excludes 5 from the set of possible values.All types of inequality symbols can be used to define different solution sets. Understanding these symbols helps in correctly identifying which values are part of the range and ensures that the solution set is accurately represented on a number line.

Plotting Inequalities

Plotting inequalities on a number line involves representing the solution set visually. This process includes:- Identifying the correct solution set using inequality symbols.- Placing closed or open circles at boundary points depending on the symbols.For the inequality \(-2 \leq x < 5\):- Place a closed circle at -2 to indicate that -2 is included in the solution set.- Place an open circle at 5 to show that 5 is not included.Then, shade or draw a line segment between these two points. This shading indicates all the values \(x\) can take. By plotting inequalities, we can visually interpret which values satisfy the inequality conditions, making it a valuable technique in algebra.