Problem 22 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(\frac{3 t+6}{2}<3 t+6\)

Step-by-Step Solution

Answer

The solution is \(t > -2\). Graph with an open circle at -2 and shade to the right.

Step 1: Eliminate the Fraction

First, eliminate the fraction by multiplying both sides of the inequality by 2 to simplify the expression, resulting in the inequality \(3t + 6 < 2(3t + 6)\).

Step 2: Distribute and Simplify

Distribute the 2 on the right side: \(2(3t + 6) = 6t + 12\). This gives us the inequality: \(3t + 6 < 6t + 12\).

Step 3: Move Terms Involving t

Subtract \(3t\) from both sides to get all terms involving \(t\) on one side: \(6 < 3t + 12\).

Step 4: Isolate the Variable Term

Subtract 12 from both sides to isolate the \(t\) term: \(-6 < 3t\).

Step 5: Solve for t

Divide both sides by 3 to solve for \(t\): \(-2 < t\). This can also be written as \(t > -2\).

Step 6: Graph the Solution on a Number Line

To graph \(t > -2\) on a number line, draw a number line and place an open circle at -2 to indicate that -2 is not included. Shade the line to the right of -2 to show that \(t\) is greater than -2.

Key Concepts

Solving Inequalities

Solving inequalities is a fundamental part of algebra that involves finding the set of values that satisfy an inequality. An inequality compares two expressions, showing whether one is less than, greater than, or equal to another, except with some flexibility compared to a simple equation.

For instance, in the inequality \( \frac{3t + 6}{2} < 3t + 6 \), our goal is to determine all possible values of \( t \) that make this statement true, rather than finding a single solution.

The process is similar to solving equations, but with added considerations, particularly when multiplying or dividing by negative numbers—which requires flipping the inequality sign. Understanding how to handle inequalities gives students the power to handle a wide range of real-world problems, from simple comparisons to complex scenarios.

Number Line Graph

A number line graph is a visual representation used to illustrate the solution set of an inequality. It helps us see the range of values that satisfy the inequality at a glance.

For the inequality \( t > -2 \), we need to draw a number line and indicate the solution set.

• First, plot an open circle at -2. This shows that -2 itself is not included in the solution set because \( t \) is strictly greater than -2.
• Next, shade the region to the right of -2 to indicate that all numbers greater than -2 are part of the solution.

Number line graphs are essential tools in inequalities, providing an intuitive way to understand and communicate ranges of solutions.

Algebraic Manipulation

Algebraic manipulation involves transforming mathematical expressions to simplify or solve them. When solving inequalities, this might include operations such as distributing, combining like terms, and factoring.

In the problem \( 3t + 6 < 6t + 12 \), algebraic manipulation came into play:

• Distributing 2 gave \( 6t + 12 \).
• Subtracting \( 3t \) from both sides helped consolidate \( t \)-terms to one side of the inequality.
• Subtracting 12 from both sides facilitated isolation of the variable term.

The ability to manipulate expressions algebraically is a powerful skill, opening pathways to not only solving inequalities but also tackling more advanced mathematical challenges.

Variable Isolation

Variable isolation is the process of re-arranging an equation or inequality so that the variable of interest is alone on one side. This step is crucial in pinpointing the solution set for inequalities.

In the exercise, isolating \( t \) was key after moving terms around.

• First, by subtracting 12 and obtaining \(-6 < 3t\).
• Then, dividing both sides by 3 simplified the inequality to \(-2 < t\), clearly indicating that \( t \) must be greater than \(-2\).

Variable isolation distills complex expressions into simple statements, making problem-solving straightforward and effective. Understanding this step enhances one's ability to approach algebraic inequalities and equations with confidence.