Problem 7 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$-3(b-1)>18$$

Step-by-Step Solution

Answer

The solution is \(b < -5\); graph with an open circle at \(-5\) and shaded to the left.

Step 1: Distribute the Negative Sign

The inequality given is \(-3(b-1) > 18\). The first step is to distribute the \(-3\) across the parentheses. Multiply \(-3\) with each term inside the parentheses: \(-3 imes b\) and \(-3 imes -1\). This results in the expression \(-3b + 3\).

Step 2: Rewrite Inequality

After distribution, the inequality becomes \(-3b + 3 > 18\).

Step 3: Isolate the Variable Term

Subtract 3 from both sides of the inequality to isolate the \(-3b\) term on the left side: \(-3b + 3 - 3 > 18 - 3\). This simplifies to \(-3b > 15\).

Step 4: Solve for 'b'

Now solve for \(b\) by dividing both sides of the inequality by \(-3\). Remember to reverse the inequality sign when dividing by a negative number: \(b < -5\).

Step 5: Check the Solution

Select a number smaller than \(-5\) to check if it satisfies the original inequality. Let's try \(b = -6\):Substitute into the original inequality: \(-3(-6 - 1) > 18\) becomes \(-3(-7) > 18\), which simplifies to \(21 > 18\). This is true, so \(b < -5\) is the correct solution.

Step 6: Graph the Solution

To graph \(b < -5\), draw a number line, place an open circle on \(-5\), and shade all the area to the left of \(-5\) to represent all numbers less than \(-5\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that enables us to eliminate parentheses by distributing a multiplication over addition or subtraction inside the parentheses. This approach simplifies complex expressions and is key to solving equations and inequalities efficiently. In the example

• Given inequality: \(-3(b-1) > 18\)
• Distribute \(-3\) across each term inside the parentheses: \(-3 \times b\) and \(-3 \times (-1)\)

This results in the simplified expression \(-3b + 3\). If you visualize, this process is like spreading the operation (here multiplication) over each term within the parentheses. Remember, distributing a negative number across the parentheses means dealing with both the sign and the magnitude properly—which requires multiplying the signs as well. This is essential for preserving the equality or inequality you are working with.

Graphing Inequalities

Graphing inequalities is a powerful way to visually represent solutions to inequalities on a number line. By graphing, we can see all possible solutions in one glance. To illustrate, let's graph the inequality \(b < -5\):

• Draw a horizontal number line.
• Locate the number \(-5\) on this line.
• Place an open circle at \(-5\) to show that \(-5\) itself is not included in the solution.
• Shade the line extending to the left of \(-5\) because it includes all numbers less than \(-5\).

This graph answers the inequality visually, providing a clear distinction of values that satisfy \(b < -5\). Remember, an open circle signifies that the boundary number is not part of the solution set, while a closed circle means it is included.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that satisfy the inequality. It's similar to solving equations, but with special rules for operations involving inequalities.

• Start by simplifying both sides, if necessary, just like equations.
• Perform inverse operations to isolate the variable on one side of the inequality. This might involve addition, subtraction, multiplication, or division.
• Importantly, remember if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

In our case after simplification, we had \(-3b + 3 > 18\). To isolate \(b\), we subtracted \(3\) from both sides to yield \(-3b > 15\). After dividing both sides by \(-3\), the inequality sign flipped, resulting in \(b < -5\). Checking the solution by substituting a number into the original inequality, like \(-6\), confirms it holds true, further verifying that the solution is correct. This thorough approach ensures you handle inequalities accurately.