Problem 31 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$-19>\frac{y}{-0.3}$$

Step-by-Step Solution

Answer

The solution is \(y > 5.7\). Graph: Open circle at 5.7, shade to the right.

Step 1: Solve the Inequality

To solve the inequality \(-19>\frac{y}{-0.3}\), we need to isolate \(y\). Start by multiplying both sides of the inequality by \(-0.3\). Remember that multiplying an inequality by a negative number reverses the inequality sign. Multiply both sides:\(-19 \times (-0.3) < y\)Calculate the left side:\(5.7 < y\). This inequality can also be rewritten for clarity as:\(y > 5.7\).

Step 2: Check the Solution

To verify the solution \(y > 5.7\), choose a value of \(y\) greater than 5.7, say \(y = 6\), and substitute it back into the original inequality: \(-19 > \frac{6}{-0.3}\)Calculate the right side:\(\frac{6}{-0.3} = -20\)So the inequality becomes:\(-19 > -20\), which is true, confirming that our solution \(y > 5.7\) is correct.

Step 3: Graph the Solution

To graph \(y > 5.7\) on a number line, draw a number line and mark the point at 5.7. Since \(y > 5.7\), we use an open circle at 5.7 and shade the line to the right of 5.7 to indicate all numbers greater than 5.7.

Key Concepts

Reverse Inequality Sign

When solving inequalities, the direction of the inequality sign can change under certain operations. This is particularly true when you multiply or divide both sides of an inequality by a negative number.
Consider the inequality \(-19 > \frac{y}{-0.3}\): to isolate \(y\), multiply both sides by \(-0.3\). However, doing so requires reversing the inequality sign due to the rules of inequalities. The inequality becomes \(5.7 < y\) or equivalently \(y > 5.7\).
This flipping of the inequality sign is crucial to capture the fact that the relationship between the quantities changes direction when multiplied by a negative.

• Multiplication or division by a negative value reverses the inequality sign.
• A flipped inequality reflects the opposite relationship when a negative scalar is involved.

Graphing Solutions

Graphing the solution of an inequality helps to visually understand the range of values it encompasses. For the inequality \(y > 5.7\), the goal is to represent all values of \(y\) that satisfy this condition.
A simple way to do this is by sketching or imagining it on a number line. You mark the number 5.7 with an open circle. This circle symbolizes that 5.7 itself is not included in the solution set.

• Use red for value-exclusions like open circles.
• Solutions greater than 5.7 extend to the right on the number line.

Shading to the right of the open circle portrays the fact that all values larger than 5.7 satisfy the inequality \(y > 5.7\). This visual method is quite intuitive and makes it easier to compare different ranges of solutions.

Number Line Representation

The number line representation of an inequality is a powerful visual tool. It shows which segments of the number line are included in the solution set.
For instance, representing the inequality \(y > 5.7\) involves first marking the value 5.7 on the number line. Since the inequality does not include 5.7 (as indicated by 'greater than' without 'or equal to'), use an open circle.

• An open circle indicates the exclusion of the number at the point it is placed.
• Shading direction indicates range—right for greater than, left for less than.

This method is particularly useful for algebra students to determine how solutions align, overlap, or remain distinct from other potential intervals and numbers, providing a clear and immediate understanding of which values satisfy the inequality.