Problem 25 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$t+7>12$$

Step-by-Step Solution

Answer

The solution is \(t > 5\). Graph: open circle at 5, shade right.

Step 1: Isolate the Variable

The given inequality is \(t + 7 > 12\). To isolate \(t\), subtract 7 from both sides of the inequality. This operation will help us find the expression of \(t\) on its own. \[t + 7 - 7 > 12 - 7\]This simplifies to:\[t > 5\]

Step 2: Graph the Solution

To graph the solution \(t > 5\) on a number line, first draw a number line and locate the point 5 on it. Since \(t\) is greater than 5, you will draw an open circle around the number 5 to show that 5 is not included in the solution set. Then, shade the line to the right of 5 to represent all numbers greater than 5.

Key Concepts

Prealgebra

Prealgebra serves as a foundational step in learning algebra, where we handle basic mathematical operations and prepare for more complex concepts. When we discuss prealgebra, we're usually dealing with simple equations and inequalities that set the stage for future studies in algebra. Understanding prealgebra is crucial as it helps develop problem-solving skills such as isolating a variable, applying arithmetic operations, and understanding mathematical expressions.
In the context of inequalities, prealgebra involves recognizing terms and rearranging them to make sense of how values compare to each other under specific conditions. For example, understanding why subtracting 7 from both sides in the inequality \(t + 7 > 12\) leads to \(t > 5\) is a prealgebra skill. This understanding is vital because it lays the groundwork for tackling far more intricate algebraic equations down the line.

Solving Inequalities

Solving inequalities is a fundamental concept in mathematics where the goal is to find the range of possible solutions that satisfy the inequality statement. In our example, we have the inequality \(t + 7 > 12\). The primary task is to isolate the variable on one side of the inequality.

• We start by performing the same operation on both sides of the inequality to maintain its balance. In this case, subtracting 7 from both sides simplifies the inequality to \(t > 5\).


• The solution, \(t > 5\), indicates that any number greater than 5 satisfies the original inequality.

It’s important to remember that the solution process for inequalities resembles that of equations, but with a crucial distinction. Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. However, this condition isn't part of our current example, but keeping this rule in mind will help solve complex inequalities in the future.

Graphing Inequalities

Graphing inequalities is essential because it provides a visual representation of the solution set. This visual tool helps confirm the range of values that satisfy the inequality.
To graph \(t > 5\), follow these steps:

• Start by drawing a horizontal number line and marking the point that corresponds to 5. This point serves as a crucial boundary for the inequality.
• Because the inequality is \(>\), rather than \(\ge\), we use an open circle on 5 to indicate that 5 itself is not included in the solution.


• The final step involves shading the line extending to the right of the open circle, representing all numbers greater than 5. This shaded line captures every possible solution to the inequality \(t > 5\).

This graphical method gives students a clear picture of how inequalities translate into a set of permissible values, fostering a deeper conceptual understanding.

Number Line Representation

Number line representation is a simple yet powerful mathematical tool that helps visually communicate solutions to inequalities. It provides clarity by showing the relationship between numbers in a linear fashion.
In our inequality \(t > 5\), the number line makes it easy to see which numbers satisfy the inequality.

• An open circle is placed at 5, signaling that while 5 is a boundary marker, it is not part of the solution.
• Shading or drawing a line to the right of this open circle visually represents all numbers greater than 5.

The number line not only serves as a check on the solution for inequalities but also enhances comprehension by allowing students to see where numbers fall in relation to each other.
Whether you are dealing with simple prealgebra problems or more complex algebraic expressions, mastering the number line is an invaluable skill in mathematics.