To determine the correct graph for the compound inequality [tex]\( 4x + 5 < -19 \text{ or } 7 - x < 6 \)[/tex], we need to solve each part of the inequality separately and then combine the results.
First, let's solve the inequality [tex]\( 4x + 5 < -19 \)[/tex]:
1. Subtract 5 from both sides:
[tex]\[ 4x < -19 - 5 \][/tex]
[tex]\[ 4x < -24 \][/tex]
2. Divide both sides by 4:
[tex]\[ x < -6 \][/tex]
So, the solution to [tex]\( 4x + 5 < -19 \)[/tex] is [tex]\( x < -6 \)[/tex].
Next, let's solve the inequality [tex]\( 7 - x < 6 \)[/tex]:
1. Subtract 7 from both sides:
[tex]\[ -x < 6 - 7 \][/tex]
[tex]\[ -x < -1 \][/tex]
2. Multiply both sides by -1 (remember to reverse the inequality sign when multiplying by a negative number):
[tex]\[ x > 1 \][/tex]
So, the solution to [tex]\( 7 - x < 6 \)[/tex] is [tex]\( x > 1 \)[/tex].
Combining these results, the solution to the compound inequality [tex]\( 4x + 5 < -19 \text{ or } 7 - x < 6 \)[/tex] is:
[tex]\[ x < -6 \text{ or } x > 1 \][/tex]
Now, let’s consider the graph for this inequality. The graph will have two regions:
1. A region where [tex]\( x < -6 \)[/tex] will be represented as a ray extending to the left from -6 (but not including -6, so an open circle at -6).
2. A region where [tex]\( x > 1 \)[/tex] will be represented as a ray extending to the right from 1 (but not including 1, so an open circle at 1).
Therefore, the correct graph should show two disjoint rays:
- One that starts just to the left of -6 and extends infinitely in the negative direction.
- Another that starts just to the right of 1 and extends infinitely in the positive direction.
Please identify the graph that matches this description from the provided options A, B, C, and D.
