Problem 39 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(2(g+4)<3 g-2(g-5)\)

Step-by-Step Solution

Answer

The solution is \(g < 2\), graphed with an open circle at 2, shaded to the left.

Step 1: Distribute and Simplify

First, apply the distributive property to eliminate the parentheses on both sides of the inequality. Expand the expression:On the left: \[2(g+4) = 2g + 8\]On the right: \[2(g-5) = 2g - 10\]Substitute these into the inequality:\[2g + 8 < 3g - 2g + 10\]Simplify:\[2g + 8 < g + 10\]

Step 2: Isolate the Variable

Subtract \(g\) from both sides to begin isolating \(g\):\[2g - g + 8 < 10\]Simplify:\[g + 8 < 10\]Next, subtract 8 from both sides:\[g < 2\]

Step 3: Graph the Solution

On a number line, draw an open circle at \(g = 2\) to indicate that this value is not included in the solution. Shade the line to the left of \(2\) to show all numbers less than \(2\) are solutions.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps to simplify expressions. It allows you to clear an expression of its parentheses by distributing a multiplier to each term inside the parentheses. In our original exercise, we use the distributive property to tackle the inequality with expressions like \(2(g+4)\).

Here is how it works:

• On the left side, \(2(g+4)\) becomes \(2g + 8\).
• On the right side, \(-2(g-5)\) gets distributed to become \(-2g + 10\).

Using the distributive property helps simplify the inequality and makes it easier to solve. Mastering this property allows you to handle not just inequalities, but a wide array of algebraic expressions.

Solution Set

The solution set of an inequality is the range of values that satisfy the inequality. In the case of our exercise, we simplified the inequality to \(g < 2\).

This means that any number less than 2 is a solution to the inequality. Solution sets can sometimes be straightforward, involving one continuous range of numbers, or they may consist of multiple segments, especially in complex problems.

• In this case, the solution set is all numbers \(g\) where \(g < 2\).
• We express it using interval notation as \((-\infty, 2)\).

Understanding solution sets is crucial because it tells us exactly which values make the inequality true.

Number Line

A number line is a visual tool that helps to represent numbers graphically. In inequalities, it is often used to visualize the solution set because it shows which parts of the line meet the criteria of the inequality.

When graphing \(g < 2\) on a number line:

• We draw an open circle at \(2\) to indicate that \(2\) is not part of the solution.
• Then, we shade the line to the left of \(2\) to show all numbers less than \(2\).

This visual representation makes it easier to grasp and verify the range of possible solutions. The number line helps clarify which values are included or excluded from the solution set.

Graphing Inequalities

Graphing inequalities involves showcasing the solution set on a graph or number line. This method enhances understanding by providing a clear, visual interpretation of the algebraic solution.

To graph \(g < 2\):

• We start by plotting an open circle at \(2\). An open circle means 2 is not included in the solution.
• We then shade the entire section of the line to the left of \(2\). This shading shows all the values of \(g\) that satisfy \(g < 2\).

Graphing makes it easier to interpret solutions and can be a powerful tool in understanding the relationships between numbers in inequalities. It also helps in checking the accuracy of our work by providing a straightforward visual answer.