Solve and use set notation.
a) 6+2x > 0 > -10+2x
Can someone explain how to solve this?
I also had trouble doing this question:
b) x+1 < 2x+7 < x+5
TIA !
To solve the given inequalities using set notation, we need to isolate the variable x in each inequality. Let's solve them step by step:
a) 6 + 2x > 0 > -10 + 2x
First, let's isolate x in each inequality.
6 + 2x > 0 -> Subtracting 6 on both sides: 2x > -6 -> Dividing by 2: x > -3
Similarly,
-10 + 2x > 0 -> Adding 10 on both sides: 2x > 10 -> Dividing by 2: x > 5
Now, we can express the solution using set notation. The solution is the set of all values of x that satisfy both inequalities, so we take the intersection of the two sets:
{x ∈ ℝ | x > -3} ∩ {x ∈ ℝ | x > 5}
Since x cannot be greater than both -3 and 5 simultaneously, the intersection of these two sets is empty. Therefore, there is no solution that satisfies both inequalities.
b) x + 1 < 2x + 7 < x + 5
Similarly, let's isolate x in each inequality.
x + 1 < 2x + 7 -> Subtracting x on both sides: 1 < x + 6 -> Subtracting 6: -5 < x
2x + 7 < x + 5 -> Subtracting x and 7 on both sides: x < -2
Next, using set notation, we express the solution as the set of all values of x that satisfy both inequalities:
{x ∈ ℝ | -5 < x} ∩ {x ∈ ℝ | x < -2}
The intersection of these two sets gives us the final solution:
{x ∈ ℝ | -5 < x < -2}