Question

Solve the inequality and graph its solution. \(-2>b-5\)

Short answer

The solution to the inequality \(-2>b-5\) is \(-\infty, 3)\ represented on the number line with a hollow dot at 3 and an arrow pointing towards lesser values.

Step-by-step solution

Isolate the Variable

To isolate 'b', add the same value to both sides of the inequality that makes the coefficient of 'b' on the right side of the inequation zero. In this case, that's 5. Hence, \[-2 + 5 > b-5 + 5\]. This simplifies to \[3 > b\].

Interpret the Inequality

The inequality \[3 > b\] means all numbers less than 3, but not including 3, making the solution set \(-\infty, 3)\ open interval.

Draw the Graph

Place a number line. Mark a hollow dot at 3 (because 3 is not included in the solution set), and an arrow pointing left, towards lesser values, starting from this point to indicate all numbers less than 3 are part of the solution set.

Key concepts

Inequality Graphing

When tasked with solving inequalities, it's crucial to represent the solution visually. Inequality graphing involves plotting the range of solutions on a number line, which provides a clear and immediate understanding of all possible values that satisfy the inequality.

Let's take the inequality \( -2 > b - 5 \) as an example. After solving for 'b' and determining that \( b < 3 \) is the solution, we move on to graphing. Using a number line is the best approach; you mark a point on the line where the number 3 would be. Since the number 3 itself is not included in the solution (often referred to as an 'open interval'), you'd use a hollow dot or circle to represent this on the graph. Then, to indicate that every number less than 3 is a solution, you draw an arrow that extends to the left from the hollow dot towards negative infinity.

This visual component helps solidify the concept of which numbers are solutions to the inequality, which is particularly helpful when dealing with compound inequalities or ones involving absolute values.

Variable Isolation

Variable Isolation is a fundamental step in solving inequalities, as it aims to get the variable of interest alone on one side of the inequality sign, allowing us to clearly see the set of solutions.

In the given example, \( -2 > b - 5 \), we add 5 to both sides to isolate 'b'. This move ensures that whatever operation we perform on one side of the inequality, we do the same to the other to maintain equality. This manipulation results in \( b < 3 \) after simplification.

When isolating the variable, remember to reverse the inequality sign if you multiply or divide both sides by a negative number. This step is often overlooked, yet it's pivotal in reaching the correct solution. For students, visualizing each step can aid comprehension—consider illustrating this with number lines or balance scales in educational content.

Inequality Notation

Inequality Notation is another significant element. It's the language we use to describe the relationship between quantities. Understanding the symbols and how they convey open or closed intervals — whether certain values are included or excluded in the solution set — is key.

In the example \( b < 3 \), the '<' symbol indicates that 'b' is less than 3, but does not include 3 itself, which we refer to as an open interval. This is notated in interval notation as \( (-\infty, 3) \), underscoring that 3 is the upper bound of the solutions but is itself not a solution. Alternatively, if 'b' had been less than or equal to 3, this would be a closed interval, including 3, represented as \( (-\infty, 3] \).

Using the correct notation ensures clear communication and prevents misunderstandings in the realm of mathematics. Accompanied by a graph, it allows students to double-check their solutions and to become fluent in the syntax of inequality.