Choose The Correct Solution And Graph For The Inequality.

Robert Capa

2 min read

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27-11-2024

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Solving inequalities involves finding the range of values that satisfy a given mathematical expression. This often requires understanding the properties of inequalities and how they behave under different operations. Let's break down how to choose the correct solution and its corresponding graph.

Understanding Inequalities

Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare expressions. Solving an inequality means isolating the variable to determine the values that make the inequality true.

Key Properties

• Adding or Subtracting: Adding or subtracting the same number from both sides of an inequality does not change the inequality's direction.
• Multiplying or Dividing by a Positive Number: Multiplying or dividing both sides by a positive number does not change the inequality's direction.
• Multiplying or Dividing by a Negative Number: Crucially, multiplying or dividing both sides by a negative number reverses the inequality's direction. For example, x < 2 becomes x > -2 if you multiply both sides by -1.

Steps to Solving an Inequality

1. Simplify Both Sides: Combine like terms and simplify the expressions on both sides of the inequality.

2. Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember to apply the rules mentioned above concerning negative numbers.

3. Determine the Solution Set: The solution set is the range of values that satisfy the inequality. This will often be expressed in interval notation (e.g., (-∞, 5] meaning all values less than or equal to 5) or set-builder notation (e.g., {x | x ≤ 5}).

4. Graph the Solution: Represent the solution set graphically on a number line. Use open circles (◦) for < and > (values not included) and closed circles (•) for ≤ and ≥ (values included). Shade the region representing the solution set.

Example

Let's solve the inequality: 3x + 5 < 11

1. Simplify: The inequality is already simplified.

2. Isolate x:

   • Subtract 5 from both sides: 3x < 6
   • Divide both sides by 3: x < 2

3. Solution Set: The solution set is all values of x less than 2. In interval notation, this is (-∞, 2).

4. Graph: The graph would show a number line with an open circle at 2 and the region to the left of 2 shaded.

Choosing the Correct Graph

When presented with multiple graphs, carefully examine the following:

• Type of Circle: Is it an open circle (◦) or a closed circle (•)? This indicates whether the endpoint is included in the solution.
• Shaded Region: Is the region to the left or right of the endpoint shaded? This corresponds to the direction of the inequality.

By following these steps and understanding the properties of inequalities, you can confidently choose the correct solution and its corresponding graph. Remember to always double-check your work to ensure accuracy.