Problem 9 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$x+3.75 \leq 5$$

Step-by-Step Solution

Answer

\(x \leq 1.25\); graph includes a filled dot at 1.25, extending left.

Step 1: Isolate 'x'

To solve the inequality \(x + 3.75 \leq 5\), we need to isolate \(x\) on one side of the inequality. Let's subtract 3.75 from both sides of the inequality to achieve this. This gives us:\[x + 3.75 - 3.75 \leq 5 - 3.75\]Simplifying both sides, we get:\[x \leq 1.25\]

Step 2: Graph the Solution

Now that we have \(x \leq 1.25\), we can represent this on a number line. To do this, plot a filled or solid dot at 1.25 to show that 1.25 is included in the solution. Then, draw a line extending to the left from the dot, indicating all values less than 1.25 are included in the solution set.

Key Concepts

Solving Inequalities

When solving inequalities, the goal is to find the range of values that satisfy the given statement. In our example, we need to solve the inequality \(x + 3.75 \leq 5\). The key to solving inequalities is to isolate the variable, which in this case is \(x\). This is similar to solving equations, but be careful with inequality signs. If you ever multiply or divide by a negative number, remember to flip the inequality sign. For this exercise, subtracting 3.75 from both sides isolates \(x\). Doing the math gives \(x \leq 1.25\), indicating that \(x\) can be any number 1.25 or smaller. This is the solution to the inequality.

Graphing Inequalities

Graphing inequalities involves showing all the possible solutions on a number line. Once you have the solved inequality, like \(x \leq 1.25\), it's time to represent it visually. On a number line, use a solid or filled dot at 1.25. This tells us that 1.25 is part of the solution, as the inequality includes usage of 'less than or equal to'. Then, draw a line stretching to the left from the dot, highlighting that all numbers less than 1.25 satisfy the inequality. This visual representation helps clarify the range of solutions, making it easier to understand the breadth of what satisfies the inequality.

Prealgebra

Prealgebra forms the foundation for understanding algebra, and dealing with inequalities is a crucial part. In prealgebra, students learn to handle basic operations, such as addition and subtraction, which are essential when manipulating inequalities. Understanding how to isolate a variable in an inequality lays the groundwork for more complex algebraic problems. It's important to grasp these basics firmly as they play a significant role in solving and interpreting more advanced mathematical expressions and problems. This exercise is a typical example where prealgebra skills come into play, simplifying the process of solving inequalities effectively.

Number Line Representation

Representing solutions on a number line helps make abstract mathematical concepts more tangible. For inequalities, this is especially useful as it clearly delineates the range of possible solutions. Using our example, plotting \(x \leq 1.25\) requires:

• Placing a filled dot on 1.25 on the number line to show it is included in the solution.
• Drawing a line extending left from 1.25 to show all numbers smaller than it are part of the solution.

This visual tool is not only easy to understand but also helps in checking solutions and communicating them to others. It's a practical approach that supports clear and straightforward mathematical communication.