Question
Solve and graph the compound inequality on a number line.
x/4 < -1 and x + 6 > -2
1. {all real numbers}
2. -6 < x < 6
3. x > -6
4. -8 < x < -4
x/4 < -1 and x + 6 > -2
1. {all real numbers}
2. -6 < x < 6
3. x > -6
4. -8 < x < -4
Answers
GPT 3.5
To solve the compound inequality, we will solve each inequality separately and then find the intersection of the solutions.
First inequality: x/4 < -1
To isolate x, we can multiply both sides of the inequality by 4 (keeping in mind that multiplying by a negative number reverses the inequality):
x < -4
Second inequality: x + 6 > -2
To isolate x, we can subtract 6 from both sides of the inequality:
x > -8
Thus, we have x < -4 and x > -8 as the solutions to the two inequalities.
Now, let's find the intersection of these two solutions. From the first inequality, we have x < -4, and from the second inequality, we have x > -8. Therefore, the intersection of these two solutions is the range where x satisfies both inequalities, which is -8 < x < -4.
Graphically, this can be represented on a number line as:
----(-8)----(-4)---->
The shaded region on the number line represents the solution to the compound inequality: -8 < x < -4. Therefore, the correct answer is option 4: -8 < x < -4.
First inequality: x/4 < -1
To isolate x, we can multiply both sides of the inequality by 4 (keeping in mind that multiplying by a negative number reverses the inequality):
x < -4
Second inequality: x + 6 > -2
To isolate x, we can subtract 6 from both sides of the inequality:
x > -8
Thus, we have x < -4 and x > -8 as the solutions to the two inequalities.
Now, let's find the intersection of these two solutions. From the first inequality, we have x < -4, and from the second inequality, we have x > -8. Therefore, the intersection of these two solutions is the range where x satisfies both inequalities, which is -8 < x < -4.
Graphically, this can be represented on a number line as:
----(-8)----(-4)---->
The shaded region on the number line represents the solution to the compound inequality: -8 < x < -4. Therefore, the correct answer is option 4: -8 < x < -4.