Problem 82 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. \((\text {Lesson } 6.1)\) $$ 6 \geq x+1 $$

Step-by-Step Solution

Answer

The solution to the inequality \(6 \geq x+1 \) is \(x \leq 5 \). The graphical representation will have an arrow extending towards the left from the point 5.

Step 1: Isolation of Variable

Start by subtracting 1 from both sides to isolate \(x\) on one side. This gives: \(x \leq 6 - 1\)

Step 2: Solving the Inequality

Simplify the inequality to get: \(x \leq 5\)

Step 3: Graphing the Solution

To graph the solution set, draw a number line and mark the point corresponding to 5. Given that \(x\) is less than or equal to 5, the arrow extends towards the left with a filled circle at 5 indicating that 5 is included in the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential skill for visualizing solutions to problems when dealing with unknown values. Let's break down the steps needed to successfully graph inequalities.
First, ensure that your inequality is simplified and variables are isolated (we'll get to that later). This will make graphing much easier.
Next, use a number line to visualize the inequality. The number line is a simple but powerful tool that helps in understanding the range of possible solutions.
For instance, consider the inequality \(x \leq 5\). On a number line, you'd place a filled circle on the number 5. This is because the "equal to" part of \(\leq\) indicates that 5 is a part of the solution. After placing the circle, draw an arrow extending leftwards to indicate that all numbers less than 5 are also part of the solution.

• The filled or open circle is crucial: Use a filled circle if the variable can equal the boundary point (\(\leq\) or \(\geq\)), indicating that the boundary number is included in the solution set.
• Use an open circle for strict inequalities, like \(<\) or \(>\), showing that the boundary number is not a solution.

By practicing these steps, one can easily master graphing inequalities, providing a clearer understanding of the range of solutions. Consistency in visual representation helps in approaching even more complex mathematical problems.

Isolation of Variable

The isolation of a variable is a key concept in algebra. This process involves rearranging an equation or inequality such that the variable of interest is by itself on one side. This makes it easier to understand and solve the relation.
In our inequality example, \(6 \geq x + 1\), our goal is to isolate the variable \(x\) on one side. Here's how to effectively do that:
Subtract the constant (in this case, 1) on the same side as the variable. So, subtracting 1 from both sides gives \(6 - 1 \geq x\). Simplifying this gives \(5 \geq x\), which is equivalent to \(x \leq 5\). This step ensures that you clearly see the condition under which \(x\) satisfies the inequality.

• Always perform the same operation on both sides of the inequality to maintain balance.
• Remember that if you multiply or divide by a negative number, the inequality sign flips direction.

Through the isolation of variables, solving inequalities becomes a systematic process. This foundational skill supports further understanding in more advanced algebraic concepts.

Number Line Representation

Number line representation is a visual approach to expressing the set of solutions for an inequality.
A number line acts like a ruler, showing numbers in sequence, where each point represents a real number. This tool helps in easily identifying the scope of solutions an inequality encompasses.
For the inequality \(x \leq 5\), the number line helps show the extent of values \(x\) can take:

• First, locate and mark the number 5 on the number line with a filled circle, indicating it is included in the solution (as per \(\leq\)).
• Second, draw an arrow extending to the left from 5, showing that all numbers less than 5 satisfy the inequality.

This filled circle and leftward arrow are quite effective in conveying the solution quickly and accurately. Graphically, it gives a precise and visual way to understand inequalities. The number line is not just limited to showing solutions but also helps in comparing and understanding relationships between numbers. Its simplicity is why it remains a fundamental tool in the study of mathematics.