Solve and graph the compound inequality on a number line.
x4 < −1 and x + 6 > −2
(1 point)
To solve and graph the compound inequality x + 4 < -1 and x + 6 > -2:
Step 1: Solve the first inequality, x + 4 < -1:
Subtract 4 from both sides of the inequality:
x + 4 - 4 < -1 - 4
x < -5
Step 2: Solve the second inequality, x + 6 > -2:
Subtract 6 from both sides of the inequality:
x + 6 - 6 > -2 - 6
x > -8
Therefore, the solution to the compound inequality is -8 < x < -5.
To graph this compound inequality on a number line:
Step 1: Draw a number line and label it accordingly.
Step 2: Mark -8 and -5 on the number line with open circles.
Step 3: Draw a solid line between the two open circles to represent all the values of x that satisfy the inequality.
Step 4: Shade the region between -8 and -5 to represent the solution set.
The graph should show an open circle at -8, an open circle at -5, and the region between -8 and -5 shaded.
To solve the compound inequality, we'll solve each inequality separately and then consider the overlap between the two solutions.
First, let's solve the first inequality:
x + 4 < -1
Subtracting 4 from both sides, we get:
x < -1 - 4
x < -5
Next, let's solve the second inequality:
x + 6 > -2
Subtracting 6 from both sides, we get:
x > -2 - 6
x > -8
Now, let's consider the overlap between the two solutions:
For x < -5 and x > -8, the only values that satisfy both inequalities are x > -5 and x > -8.
Graphing this on a number line, we'll shade the region to the right of -5 and -8 (as we need x to be greater than both):
-------------(=|----------------------------|===>
-8 -5
In interval notation, the solution is:
(-5, ∞)
To solve and graph the compound inequality on a number line, we need to break it down into two separate inequalities and then graph the solution set for each inequality on the same number line.
First, let's solve the inequality x + 4 < -1:
1. Subtract 4 from both sides of the inequality:
x + 4 - 4 < -1 - 4
Simplifying:
x < -5
Next, let's solve the inequality x + 6 > -2:
1. Subtract 6 from both sides of the inequality:
x + 6 - 6 > -2 - 6
Simplifying:
x > -8
Now, we can graph these two inequalities on the same number line:
We start by drawing a number line and marking -8 and -5 on it. Since the variable x is greater than -8, we draw an open circle on -8. Since x is less than -5, we draw an open circle on -5.
Finally, we draw an arrow from the open circle at -8 to the right, indicating that the solution set is all numbers greater than -8. We draw another arrow from the open circle at -5 to the left, indicating that the solution set is all numbers less than -5.
The resulting graph on the number line would look like this:
-8 -------------------------------> (+infinity)
o
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(-infinity) <------------------------ -5
o
So, the solution to the compound inequality is x > -8 and x < -5.