Problem 20 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(2(4 t+9) \leq 18\)

Step-by-Step Solution

Answer

The solution is \( t \leq 0 \); graph it with a closed circle at 0, shading left.

Step 1: Expand the Inequality

First, distribute 2 into the terms inside the parentheses. This gives us: \[ 2 \times 4t + 2 \times 9 \leq 18 \]which simplifies to:\[ 8t + 18 \leq 18 \]

Step 2: Isolate the Variable Term

To move the constant to the right side of the inequality, subtract 18 from both sides:\[ 8t + 18 - 18 \leq 18 - 18 \]This simplifies to:\[ 8t \leq 0 \]

Step 3: Solve for the Variable

Divide both sides of the inequality by 8 to solve for \( t \):\[ \frac{8t}{8} \leq \frac{0}{8} \]This results in:\[ t \leq 0 \]

Step 4: Graph the Solution Set on a Number Line

On a number line, draw a closed circle at 0 to indicate that 0 is included in the solution. Shade the line to the left of 0 to represent all the values less than or equal to 0.

Key Concepts

Graphing Inequalities

Inequalities are like equations, but instead of an equal sign, they use symbols such as ">", "<", "≤", and "≥". When graphing inequalities, we visually represent the solution set.
The solution set is the range of values that satisfy the inequality. Let's walk through an example to see how this is done. For the inequality \( t \leq 0 \), we first identify 0 on the number line. We then draw a solid circle at 0 to show that 0 is part of the solution.
Next, we shade the region left of the 0 because we're considering all values of \( t \) that are "less than or equal to" zero. This shading visually represents the entire solution set, making it easy to understand which values of \( t \) satisfy the inequality.

Number Line Representation

The number line is a powerful visual tool for representing the solution to an inequality. It helps in understanding which values satisfy the given condition. In a number line, each point corresponds to a number.
When plotting inequalities, use open or closed circles. A closed circle on a number indicates that this number is included in the solution set. An open circle, however, shows that the number is not part of the solution.

• Closed Circle: Used for "≤" or "≥".
• Open Circle: Used for "<" or ">".

After placing the circle, shade the line in the correct direction to indicate all possible solutions. For \( t \leq 0 \), since we have a closed circle, 0 is part of our solution set.
We shade to the left, which represents all the numbers that are less than or equal to 0.

Variable Isolation

Isolating the variable is a crucial step in solving inequalities. It's about getting the variable alone on one side of the inequality sign.
Let's break it down using our example: \( 8t + 18 \leq 18 \). The primary objective here is to "isolate" or "detach" \( t \) from other numbers.
Start by moving the constant term, 18, from the left to the right. Do this by subtraction, making sure whatever you do to one side of the inequality, you also do to the other.
This operation gives us \( 8t \leq 0 \). The variable is still multiplied by 8, so our next step is to divide each side by 8. This action gives us \( t \leq 0 \).
Always remember, when you multiply or divide each side of an inequality by a negative number, flip the inequality sign!

Inequality Solutions

Inequalities have solutions that express a range of possible values, rather than a single number. In the case of \( t \leq 0 \), the inequality tells us that \( t \) can be any number less than or equal to 0.
The solution for \( t \) isn't just one value but includes an infinite set of possibilities because numbers greater, lesser, or equal to 0 can fit into this solution framework.
Here are key points to remember when dealing with inequality solutions:

• A solution can be written with the inequality itself, such as \( t \leq 0 \), to specify the range.
• Solutions employ interval notations, such as \((-\infty, 0] \), to represent the starting and ending points.

Understanding these concepts ensures a solid grasp of how inequalities work, opening up further mathematical comprehension and problem-solving skills.