Problem 24 - Prealgebra

Question

Solve each inequality. Check your solution. Then graph the solution on a number line. $$18>-2 g$$

Step-by-Step Solution

Answer

The solution is \(g > -9\), and it is shown on the number line from \(-9\) to infinity with an open circle at \(-9\).

Step 1: Isolate the Variable

We start by isolating the variable on one side of the inequality. The given inequality is \(18 > -2g\). To isolate \(g\), we divide both sides by \(-2\). Remember that when you divide or multiply both sides of an inequality by a negative number, the inequality sign flips. Thus, \(18/-2 < g\).

Step 2: Simplify the Expression

Now, simplify \(18/-2\). Dividing, we get \(-9\). So the inequality becomes \(-9 < g\). This can also be expressed as \(g > -9\) by rewriting it.

Step 3: Check the Solution

Pick a value for \(g\) that is greater than \(-9\), such as \(0\), and substitute it back into the original inequality to verify. The original inequality is \(18 > -2g\). Substituting \(g = 0\), we have \(18 > 0\), which is true. Thus the solution holds.

Step 4: Graph on a Number Line

On a number line, represent the solution \(g > -9\) by drawing an open circle at \(-9\) to indicate \(-9\) is not included, and shade the line to the right of \(-9\) to show all numbers greater than \(-9\) are included in the solution.

Key Concepts

Graphing Inequalities on a Number Line

Understanding how to graph inequalities on a number line is an essential skill in mathematics. In our example, the inequality is given as \( g > -9 \). To represent this visually, we need to create a number line. Start by finding the point corresponding to \(-9\) on the number line. Since the inequality specifies \( g > -9 \) and not \( g \geq -9 \), we use an open circle at \(-9\). This open circle indicates that \(-9\) is not part of the solution. Then, shade the number line to the right of \(-9\) to show that all numbers greater than \(-9\) satisfy the inequality. This shaded area represents all possible values of \( g \) that make the inequality true. Graphing inequalities in this manner provides a visual representation of the range of possible solutions.

Negative Numbers in Inequalities

Handling negative numbers in inequalities requires special attention, particularly concerning the direction of the inequality sign. In the example \( 18 > -2g \), to isolate \( g \), we must divide both sides by \(-2\). However, a crucial rule of inequalities is that when you multiply or divide both sides by a negative number, you must flip the inequality sign.
Otherwise, the solution would be incorrect. Therefore, dividing \( 18 \) by \(-2\), the expression \( 18/-2 < g \) simplifies, flipping the \( > \) sign to \( < \). This yields the inequality \(-9 < g\), which is also written as \( g > -9 \).
By remembering to flip the sign, we ensure that our solution remains accurate. It's a key rule that holds for all equations and inequalities involving negative numbers. This is one of those little rules that is easy to forget but can lead to drastically different solutions if not applied.

Checking Solutions in Inequalities

Verifying a solution in inequalities reinforces the accuracy of your results and ensures understanding. After finding that \( g > -9 \), it is recommended to choose a test value greater than \(-9\) to substitute back into the original inequality.
For example, if you pick \( g = 0 \), substitute it into the original \( 18 > -2g\). This gives us \( 18 > 0 \), which is indeed true. Testing a value provides proof that the inequality holds true for values greater than \(-9\).
This step not only verifies the solution but also sheds light on the integrity of each transformation made during the solving process. Thus, checking solutions in inequalities serves as a vital part of solving problems, giving students the assurance that their approach and their answer are both correct.