Can someone shwo me how to solve these inequalties for me?
-6 < x and 3 < 6 (Remember this is an "and" statement)
-3x < -6 or x + 5 < -2
x - 2 > 2x + 1 or -10 > -2x - 2
Sure! I can help you solve these inequalities step by step.
Let's start with the first set of inequalities:
-6 < x and 3 < 6
To solve an "and" statement, we need to find the values of x that satisfy both inequalities simultaneously.
For the first inequality, -6 < x, it means that x is greater than -6. We can represent this graphically as a number line with an open circle on -6 and the arrow pointing towards positive infinity (since x can take any value greater than -6).
For the second inequality, 3 < 6, it means that 3 is less than 6. Since this inequality is always true, we don't need to represent it graphically.
So, to find the solution that satisfies both inequalities, we look for the overlapping region on the number line. In this case, the solution is x > -6.
Moving on to the next set of inequalities:
-3x < -6 or x + 5 < -2
Let's solve each inequality separately:
For the first inequality, -3x < -6, we need to isolate x. We can do this by dividing both sides of the inequality by -3. However, note that when we divide an inequality by a negative number, the direction of the inequality sign flips. So, we have to reverse the sign.
Dividing both sides by -3 gives us x > 2.
For the second inequality, x + 5 < -2, we need to isolate x. We can do this by subtracting 5 from both sides of the inequality:
x + 5 - 5 < -2 - 5
x < -7
So the solution for the second inequality is x < -7.
Finally, let's solve the last set of inequalities:
x - 2 > 2x + 1 or -10 > -2x - 2
First, let's focus on x - 2 > 2x + 1:
To solve this inequality, we need to isolate x. We can do this by subtracting x from both sides of the inequality:
x - x - 2 > 2x - x + 1
-2 > x + 1
Now, subtract 1 from both sides:
-2 - 1 > x + 1 - 1
-3 > x
So the solution for the first inequality is x < -3.
Now let's concentrate on the second inequality:
-10 > -2x - 2
To solve this inequality, we need to isolate x. We can do this by adding 2 to both sides of the inequality:
-10 + 2 > -2x - 2 + 2
-8 > -2x
Dividing both sides by -2 (and reversing the inequality sign):
4 < x
So the solution for the second inequality is x > 4.
Therefore, the overall solution for the set of inequalities x - 2 > 2x + 1 or -10 > -2x - 2 is x < -3 or x > 4.
I hope this explanation helps! Let me know if you have any further questions.