Choose The Correct Solution And Graph For The Inequality

Julian Assange

2 min read

·

03-01-2025

6

0

Share

Solving inequalities involves finding the range of values that satisfy a given mathematical expression. This process often requires understanding of algebraic manipulation and the principles of inequality. The solution is then visually represented on a number line graph. Let's break down how to choose the correct solution and graph.

Understanding the Inequality

Before we can solve an inequality, we must first understand what it's asking. Inequalities use symbols like:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

These symbols indicate a relationship between two expressions. The goal is to find the values that make the relationship true.

Solving the Inequality

The process of solving an inequality is similar to solving an equation, with one key difference: when multiplying or dividing by a negative number, you must reverse the inequality sign. For example:

• -2x < 6 becomes x > -3 (after dividing by -2 and reversing the sign).

Let's illustrate with an example:

Solve the inequality 3x + 5 ≤ 11

1. Subtract 5 from both sides: 3x ≤ 6
2. Divide both sides by 3: x ≤ 2

Therefore, the solution is x ≤ 2. This means any value of x that is less than or equal to 2 will satisfy the original inequality.

Graphing the Solution

The solution to an inequality is typically graphed on a number line.

• For inequalities with ≤ or ≥: Use a closed circle (●) at the value to indicate that the value is included in the solution.
• For inequalities with < or >: Use an open circle (○) at the value to indicate that the value is not included in the solution.

For our example, x ≤ 2, the graph would show a closed circle at 2, with an arrow extending to the left, indicating all values less than 2 are also part of the solution.

Common Mistakes to Avoid

• Forgetting to reverse the inequality sign: This is a crucial step when multiplying or dividing by a negative number.
• Incorrectly graphing the solution: Pay close attention to whether the circle should be open or closed.
• Misinterpreting the inequality symbol: Make sure you understand the meaning of each symbol (<, >, ≤, ≥).

Conclusion

Choosing the correct solution and graph for an inequality involves careful algebraic manipulation and accurate interpretation of the inequality symbols. By following the steps outlined above and paying attention to common mistakes, you can confidently solve and graph any inequality. Remember to always check your answer by substituting a value from your solution set back into the original inequality to ensure it holds true.