Problem 18 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$-3 b+4<-2$$

Step-by-Step Solution

Answer

The solution is \(b > 2\). Graph this by shading right of 2 on a number line with an open circle at 2.

Step 1: Move Constant

Start by isolating the term with the variable. Here, we need to move the constant on the left side of the inequality, which is 4. Subtract 4 from both sides of the inequality to do so: \[-3b + 4 - 4 < -2 - 4\]This simplifies to:\[-3b < -6\]

Step 2: Divide by Coefficient

Now, we need to solve for \(b\). The coefficient of \(b\) is -3. Divide both sides of the inequality by -3. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:\[-3b / -3 > -6 / -3\]So the inequality becomes:\[b > 2\]

Step 3: Check the Solution

To verify, pick a value for \(b\) greater than 2, like \(b = 3\), and substitute it back into the original inequality:\[-3(3) + 4 < -2\]\[-9 + 4 = -5\]Since -5 is indeed less than -2, our solution holds true.

Step 4: Graph the Solution

To graph the inequality \(b > 2\), draw a number line. Place an open circle at 2 to indicate that 2 is not included in the solution, and shade the region to the right of 2 to indicate all numbers greater than 2 are part of the solution set.

Key Concepts

Solving Inequalities

Inequalities like \(-3b + 4 < -2\) are widespread in math, particularly when dealing with algebraic expressions. Solving inequalities is somewhat similar to solving equations, but with a few additional rules to keep in mind.

**Step-by-step approach:**

• Begin by isolating the term containing the variable. Here, we subtract 4 from both sides, simplifying to:\(-3b < -6\).
• Next, solve for the variable by dividing both sides by -3. Remember, dividing by a negative flips the inequality sign, changing it to:\(b > 2\).

Checking your work is crucial; substitute numbers greater than 2 to ensure they satisfy the original inequality. This verifies that your solution is correct, as illustrated. Always take special care with operations that might change the direction of the inequality sign.

Graphing Inequalities

Graphing inequalities involves transforming a range of solutions into a visual format. This process helps students easily understand the solution set for an inequality.

To graph the solution to \(b > 2\):

• Draw a number line and locate the number 2 on it.
• Place an open circle around the 2 to demonstrate that the value 2 is not included in the solution set.
• Shade or draw an arrow to the right of the open circle. This indicates all numbers greater than 2 are included in the solution.

In graphing inequalities, the open circle is crucial. It shows that the number itself isn't part of the solution (unlike a closed circle, which would indicate inclusion). Graphing is a powerful tool in visually representing solutions, helping identify them quickly and precisely.

Prealgebra Concepts

Prealgebra is an essential building block for understanding mathematics, especially when dealing with inequalities. At the heart of prealgebra is comprehension of fundamental operations and principles, such as:

• Variable manipulation: Terms like \(-3b + 4 < -2\) show negative and positive values working together in expressions.
• Reversing inequality signs: Understanding when and why signs switch directions (such as when dividing by a negative) is crucial for accuracy.
• Number line interpretation: Provides insight into the range of possible solutions, showing relationships between values on a line.

Grasping these basics prepares students for more advanced algebraic concepts. It helps them solve and understand complex equations, inequalities, and their solutions more effectively, setting the stage for future math success.