Problem 6 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(4 y+7>31\)

Step-by-Step Solution

Answer

The solution is \(y > 6\); graphed as an open circle at 6 with shading to the right.

Step 1: Isolate the Variable

To solve the inequality, the first step is to isolate the variable on one side of the inequality. Begin by subtracting 7 from both sides of the inequality: \(4y + 7 - 7 > 31 - 7\). This simplifies to: \(4y > 24\).

Step 2: Solve for the Variable

Next, solve for \(y\) by dividing both sides of the inequality by 4:\(\frac{4y}{4} > \frac{24}{4}\).This gives:\(y > 6\).

Step 3: Graph the Solution on a Number Line

To graph \(y > 6\) on a number line, draw a number line and locate the number 6. Since the inequality is \(>\), use an open circle around 6 to indicate that 6 is not part of the solution set. Shade the line to the right of 6 to show that all numbers greater than 6 satisfy the inequality.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In the exercise, you are working with the expression \[4y + 7 > 31\]This expression includes the variable "y," the number "7," and the coefficient "4" attached to "y." It represents an inequality, which is a type of expression that shows a relationship between different values where one side is not necessarily equal to the other.

When solving inequalities, a key step involves isolating the variable, which means getting "y" on one side by itself. You do this to understand what values "y" can take. In our example, subtracting 7 from both sides and then dividing everything by 4 helps us reach the simplified form: \[y > 6\]

This form tells us that "y" represents any number greater than 6.

Inequality Graphing

Graphing inequalities involves visually representing solutions on a graph, helping you easily see where a solution lies on the number line. For inequalities involving one variable, like "y," the graph usually appears as a portion of the number line.In this exercise, the inequality \[y > 6\] instructs us to graph numbers greater than 6, not including 6. The visual representation involves using an open circle to illustrate that 6 is not part of the solution.

From the point of 6 on the number line, you shade or draw an arrow extending to the right. This shaded area indicates all numbers greater than 6, showing the solution set of the inequality. This visual aid summarized the problem effectively and is particularly useful for students struggling to conceptualize abstract numerical relationships.

Number Line Representation

A number line is a simple yet powerful tool you use to represent numbers in a linear format. It's often employed to demonstrate sequences, operations, and solutions such as solutions to inequalities. For the inequality \[y > 6\], representing it on a number line means depicting all possible values of y that satisfy this condition.To do this:

• Draw a horizontal line.
• Mark a point for "6" along this line.
• Use an open circle centered at "6" because the value "6" itself is not included in the solution. An open circle signifies that the boundary value is not part of the set, distinguishing it from closed circles which indicate inclusion.
• Shade everything to the right of "6," signifying that all values greater than "6" fulfill the inequality's requirement.

This methodical visualization helps clarify the solution set and enhances comprehension of how inequalities are structured and solved.