consider the inequality 2x+3≤-3. Find the set of all integer solutions of this inequality that are also solutions of the inequality 5x-2<3.
Please help I am really confuded
2x+3≤-3
2x ≤ -6
x ≤ -3
AND
5x-2<3
5x < 5
x < 1
so -3 ≤ x < 1
looks like {-3, -2, -1, 0 }
To solve this problem, we need to find the set of all integer solutions that satisfy both inequalities: 2x + 3 ≤ -3 and 5x - 2 < 3.
Let's start by solving the first inequality, 2x + 3 ≤ -3:
1. Subtract 3 from both sides: 2x ≤ -6.
2. Divide both sides by 2: x ≤ -3.
Now, let's solve the second inequality, 5x - 2 < 3:
1. Add 2 to both sides: 5x < 5.
2. Divide both sides by 5: x < 1.
To satisfy both inequalities, we need to find the values of x that are less than 1, but also less than or equal to -3. So, we can write the solution as -∞ < x ≤ -3.
Now, let's find the set of all integer solutions within this range:
The integers that satisfy -∞ < x ≤ -3 are -∞, ..., -5, -4, -3.
Therefore, the set of all integer solutions that satisfy both inequalities is {-∞, ..., -5, -4, -3}.