Select the graph that would represent the best presentation of the solution set.
| y + 2 | > 6
To find the best representation of the solution set, we first need to solve the inequality:
| y + 2 | > 6
This inequality can be rewritten as two separate inequalities:
y + 2 > 6 OR - (y + 2) > 6
Solving the first inequality:
y + 2 > 6
y > 6 - 2
y > 4
Solving the second inequality:
- (y + 2) > 6
-y - 2 > 6
-y > 6 + 2
-y > 8
Multiply both sides by -1 and reverse the inequality symbol:
y < -8
So the solution set is y > 4 OR y < -8.
Now let's analyze the graphs:
Graph A: A number line from -10 to 10, with a solid dot at 4 and an arrow pointing to the right.
Graph B: A number line from -10 to 10, with a solid dot at -8 and an arrow pointing to the left.
Graph C: A number line from -10 to 10, with an open circle at 4 and an arrow pointing to the right, as well as an open circle at -8 and an arrow pointing to the left.
Graph D: A number line from -10 to 10, without any markings or arrows.
From these options, the best representation of the solution set would be Graph C. It accurately shows the open circles at 4 and -8 to indicate that those values are not included in the solution set since the inequality is strict (>|<) and not inclusive (≥≤). The arrows in both directions show that the solution set includes all values greater than 4 or less than -8.
To represent the solution set of the equation |y + 2| > 6 graphically, we need to plot the graph of both the inequality y + 2 > 6 and y + 2 < -6.
First, let's solve the equation y + 2 > 6:
y + 2 > 6
y > 6 - 2
y > 4
Now, let's solve the equation y + 2 < -6:
y + 2 < -6
y < -6 - 2
y < -8
So the solution set for the inequality |y + 2| > 6 is y < -8 or y > 4.
The best graph to represent this solution set would be a number line with an open circle at -8 and an open circle at 4, indicating that the values of y are not included within those points. The line between -8 and 4 should be shaded to represent all the values of y that satisfy the inequality. This graph visually represents the solution set of the equation |y + 2| > 6.
To represent the solution set of the inequality | y + 2 | > 6, we need to consider two cases: one where y + 2 is greater than 6, and another where y + 2 is less than -6. Let's break it down step by step:
1. Identify the two cases:
i) y + 2 > 6
ii) y + 2 < -6
2. Solve for y in each case:
i) y + 2 > 6
Subtract 2 from both sides: y > 4
ii) y + 2 < -6
Subtract 2 from both sides: y < -8
3. Graph the solution on a number line:
-8 <==============4==========================>
| | | |
-9 -8 -7 -6
On the number line, we mark a point at -8, indicating that y is less than -8, and another point at 4, indicating that y is greater than 4. We draw open circles at these points since the inequality is strict (i.e., not including -8 and 4).
4. Choose the correct representation:
For | y + 2 | > 6, the solution set consists of all values of y that are less than -8 or greater than 4. The best graph to represent this solution set would be a number line with an open circle at -8, an open circle at 4, and shading to the left and right of those points to indicate that y can take any value outside of that range.
Therefore, the answer is a number line with open circles at -8 and 4, and shading to the left and right.