Which Graph Represents the Solution Set for –4(1 – X) ≤ –12 + 2X?
When solving inequalities, it is important to understand how to represent the solution set graphically. In this article, we will discuss how to determine the graph that represents the solution set for the inequality –4(1 – X) ≤ –12 + 2X. We will also address some frequently asked questions related to this topic.
To begin, let’s simplify the given inequality:
–4(1 – X) ≤ –12 + 2X
Distributing the -4 on the left side of the inequality, we have:
–4 + 4X ≤ –12 + 2X
Next, let’s isolate the variables on one side of the inequality:
4X – 2X ≤ -12 + 4
2X ≤ -8
Dividing both sides of the inequality by 2, we obtain:
X ≤ -4
Now that we have found the solution, we can represent it graphically. To do this, we need to draw a number line and mark the value of -4. Since the inequality includes the less than or equal to symbol (≤), we will represent -4 with a closed dot on the number line. This indicates that -4 is included in the solution set.
Now, we shade the region to the left of -4 on the number line. This represents all the values of X that satisfy the inequality X ≤ -4. The shaded region indicates that any value less than or equal to -4 is a solution.
Q: What does the solution set represent in this inequality?
A: The solution set represents all the values of X that satisfy the given inequality. In this case, it represents all the values less than or equal to -4.
Q: Why is -4 included in the solution set?
A: The inequality includes the less than or equal to symbol (≤), which means that -4 is included in the solution set. If the inequality included only the less than symbol (<), -4 would not be included in the solution set. Q: How can I determine the shading direction on the number line? A: To determine the shading direction, you need to look at the inequality symbol. If the symbol is less than or equal to (≤) or greater than or equal to (≥), you shade in the direction of the values that satisfy the inequality. If the symbol is only less than (<) or greater than (>), you shade in the opposite direction.
Q: Can I use a different method to solve the inequality?
A: Yes, there are multiple methods to solve inequalities, such as graphing, substitution, or using algebraic manipulations. However, graphing is often the most straightforward method to represent the solution set visually.
Q: How can I verify if a value is in the solution set?
A: To verify if a value is in the solution set, substitute it into the original inequality and check if the inequality holds true. If the inequality is satisfied, the value is in the solution set; otherwise, it is not.
In conclusion, the graph that represents the solution set for the inequality –4(1 – X) ≤ –12 + 2X is a number line with a closed dot at -4 and shading to the left. The shaded region represents all the values of X that satisfy the inequality X ≤ -4. Remember to consider the inequality symbol and follow the appropriate shading direction.