Solve and graph the compound inequality on a number line.
x/4 < -1 and x + 6 > -2
To solve the compound inequality x/4 < -1 and x + 6 > -2, we will solve each inequality separately and then find the intersection of the solutions.
For the first inequality, we have x/4 < -1. To isolate x, we can multiply both sides by 4 (remembering to flip the inequality sign since we are multiplying by a negative number):
x/4 < -1
4*(x/4) > 4*(-1)
x > -4
So the solution to the first inequality is x > -4.
For the second inequality, we have x + 6 > -2. To isolate x, we can subtract 6 from both sides:
x + 6 > -2
x + 6 - 6 > -2 - 6
x > -8
So the solution to the second inequality is x > -8.
To find the intersection of the solutions, we look for values that satisfy both inequalities. Since x must be greater than -4 and also greater than -8, the solution is x > -4.
Now, let's graph this solution on a number line:
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
|---|---|---|---|---|---|---|---|---|---|
x > -4
The shaded area shows the solution to the compound inequality x/4 < -1 and x + 6 > -2, which is x > -4.
To solve and graph the compound inequality on a number line, let's break it down into two separate inequalities:
The first inequality is x/4 < -1. To solve this, we can follow these steps:
Step 1: Multiply both sides of the inequality by 4 (since the denominator is 4).
(x/4) * 4 < (-1) * 4
x < -4
So our first solution is x < -4.
The second inequality is x + 6 > -2. To solve this, we can follow these steps:
Step 1: Subtract 6 from both sides of the inequality.
x + 6 - 6 > -2 - 6
x > -8
So our second solution is x > -8.
Combining the solutions from both inequalities, we have:
x < -4 and x > -8
To graph this compound inequality on a number line, start by marking -4 and -8 on the number line. Then, shade the region between -4 and -8, excluding the endpoints (-4 and -8) because the inequality is strict (less than and greater than).
The graph would look like this:
```
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
| [----------------------------] |
```
The shaded region represents the values of x that satisfy the compound inequality x/4 < -1 and x + 6 > -2.
To solve the compound inequality x/4 < -1 and x + 6 > -2, we will solve each inequality separately and then find the intersection of the solutions.
Let's start with the first inequality: x/4 < -1.
1) Multiply both sides of the inequality by 4 to get rid of the fraction:
4 * (x/4) < -1 * 4
x < -4
Now let's move on to the second inequality: x + 6 > -2.
1) Subtract 6 from both sides of the inequality:
x + 6 - 6 > -2 - 6
x > -8
Now, to graph the compound inequality on a number line, we will represent both inequalities using open circles and solid lines.
Let's start with the first inequality, x < -4. Place an open circle on -4, and draw an arrow to the left because x is less than -4.
Next, for the second inequality, x > -8, place an open circle on -8 and draw an arrow to the right because x is greater than -8.
Now, combine the two number lines into one by overlapping them. The overlapping region represents the solution for both inequalities.
Final compound inequality graph:
```
-----o----------------------------------o------------------------>
-8 -4
```
The solution is the region between -8 and -4, excluding the endpoints (-8 and -4).