Solve and graph the compound inequality on a number line x/4 < -1 and x + 7 > -2 Select the correct answer from the following
a) x > -6
b) All real numbers
c) -6 < x < 6
d) - 8 < x < -4
Show your work
To solve the compound inequality:
x/4 < -1 and x + 7 > -2
Let's solve each inequality separately and then find the intersection of their solutions.
1) x/4 < -1
Multiply both sides of the inequality by 4 (since 4 is positive and we don't need to flip the inequality sign):
x < -4
2) x + 7 > -2
Subtract 7 from both sides of the inequality:
x > -9
Now let's find the intersection of these two solutions. We want values of x that satisfy both inequalities.
The solutions to the compound inequality are x < -4 and x > -9.
Graphically on a number line:
--------------(-9)======(-4)-----------------
The shaded area represents the values of x that satisfy both inequalities.
The correct answer is d) -8 < x < -4.
To solve and graph the compound inequality x/4 < -1 and x + 7 > -2, we need to solve each inequality separately and then find the intersection of their solution sets.
1) Solving x/4 < -1:
To isolate x, we multiply both sides of the inequality by 4 (remembering to reverse the inequality symbol since we are multiplying by a negative number):
x/4 < -1
4(x/4) > 4(-1)
x > -4
2) Solving x + 7 > -2:
To isolate x, we subtract 7 from both sides of the inequality:
x + 7 > -2
x + 7 - 7 > -2 - 7
x > -9
So the solutions to each inequality individually are x > -4 and x > -9.
Now, we need to find the intersection of these solution sets. Since both inequalities have the same direction for x, i.e., x is greater than some value, the intersection is the greater value of the two solutions. In other words, x must satisfy both inequalities, so it must be greater than -4 and -9.
The greater value between -4 and -9 is -4. Therefore, the compound inequality can be written as x > -4.
On a number line, this would be represented by an open circle at -4 and a shaded region to the right, indicating that x is greater than -4.
Answer: a) x > -6
To solve and graph the compound inequality, we will break it down into two separate inequalities and then find the intersection between the solutions. Let's start with the first inequality:
1) x/4 < -1
To solve for x, we will multiply both sides of the inequality by 4 (since we want to isolate x):
4 * (x/4) < 4 * (-1)
x < -4
So the first inequality gives us x < -4.
Now let's move on to the second inequality:
2) x + 7 > -2
To solve for x, we will subtract 7 from both sides of the inequality:
x + 7 - 7 > -2 - 7
x > -9
So the second inequality gives us x > -9.
To find the intersection between the solutions of the two inequalities, we need to find the values of x that satisfy both x < -4 and x > -9 simultaneously.
We can draw a number line and mark -4 and -9 on it:
___________(-9)_____________(-4)__________
The solution set will be the values of x that lie between -9 and -4. So, the correct answer is:
d) -8 < x < -4.