Problem 56 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. (lesson \(1-5 )\) $$ 2 d+15 \geq 3 $$

Step-by-Step Solution

Answer

The solution is \(d \geq -6\). Graphically, plot a closed circle at -6 and shade to the right.

Step 1: Subtract 15 from Both Sides

We start by isolating the term containing the variable \(d\). To do this, subtract 15 from both sides of the inequality: \[2d + 15 - 15 \geq 3 - 15\]This simplifies to:\[2d \geq -12\]

Step 2: Divide Both Sides by 2

Next, we divide both sides of the inequality by 2 to solve for \(d\): \[\frac{2d}{2} \geq \frac{-12}{2}\]This simplifies to:\[d \geq -6\]

Step 3: Graph the Solution on a Number Line

The solution to the inequality is all numbers greater than or equal to -6. On a number line, plot a closed circle on -6 to indicate that -6 is included in the solution set, and shade all numbers to the right of -6, showing all numbers greater than or equal to it.

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions that show the relationship between two values that are not equal. They use symbols like \( > \), \( < \), \( \geq \), or \( \leq \). Solving inequalities involves finding all possible values of the variable that make the inequality true.
To solve an inequality like \( 2d + 15 \geq 3 \), you should aim to have the variable on one side of the inequality. Here's how you do it:

• Isolate the Variable: Start by eliminating constant terms on the same side as the variable. In this case, subtract 15 from both sides, leading to a simpler inequality \( 2d \geq -12 \).


• Divide or Multiply: If the variable is multiplied by a coefficient (like 2d), divide both sides by this number to solve for the variable. This gives \( d \geq -6 \). Remember, if you divide or multiply by a negative number, the inequality sign flips direction.

Once you've solved the inequality, you identify the solution set, which includes all values that satisfy the inequality.

Graphing Inequalities

Graphing inequalities allows us to visually represent solutions on a number line or coordinate plane. For our inequality \( d \geq -6 \), we use a number line for a straightforward illustration.

• Identify the Boundary Point: In \( d \geq -6 \), the boundary point is -6. This point is crucial because it marks where the range of solutions starts.


• Choose the Type of Circle: Use a closed circle at -6 because the inequality includes -6 (\( \geq \) or \( \leq \)), implying that the boundary point is part of the solution set.


• Shade the Region: Near-free and to the right of the closed circle, shade the number line to show all numbers greater than or equal to -6 represent the solution for \( d \).

Graphing in this manner helps visually confirm the solution to the inequality and provides an intuitive understanding of the range of values involved.

Number Line Representation

Number line representation is a practical way to display the solution to inequalities. It provides visual clarity and makes it easy for others to see the set of solutions.
When plotting \( d \geq -6 \) on a number line:

• Create the Line: Draw a horizontal line and mark equal segments to represent numbers.


• Locate the Critical Point: Identify and mark the boundary point; here, it's -6. This point divides the number line into two potential parts.


• Use Symbols Appropriately: A closed circle is placed at -6 due to the \( \geq \) symbol, indicating inclusion of the number -6.


• Shade the Correct Direction: Shade to the right of -6, showing all numbers that are solutions to the inequality \( d \geq -6 \).

This method clearly represents solutions and helps ensure no part of the solution set is overlooked.