Solve the linear inequality. Express the solution using interval notation.
−13 < 3x + 2 ≤ 11
We will solve each part of the inequality separately.
For the first part, -13 < 3x + 2, subtract 2 from both sides:
-13 - 2 < 3x + 2 - 2
-15 < 3x
Then divide both sides by 3, remembering to reverse the inequality when dividing by a negative number:
-15/3 > 3x/3
-5 > x
So the solution to the first inequality is x < -5.
For the second part, 3x + 2 ≤ 11, subtract 2 from both sides:
3x + 2 - 2 ≤ 11 - 2
3x ≤ 9
Then divide both sides by 3:
3x/3 ≤ 9/3
x ≤ 3
So the solution to the second inequality is x ≤ 3.
In interval notation:
The solution to -13 < 3x + 2 ≤ 11 is (-∞, -5) ∪ (-∞, 3].
To solve the given linear inequality, we need to isolate the variable x in the expression 3x + 2.
First, let's solve the first part of the inequality:
-13 < 3x + 2
Subtract 2 from both sides:
-13 - 2 < 3x + 2 - 2
-15 < 3x
Now, divide both sides by 3 (since we want to isolate x):
-15/3 < 3x/3
-5 < x
Next, let's solve the second part of the inequality:
3x + 2 ≤ 11
Subtract 2 from both sides:
3x + 2 - 2 ≤ 11 - 2
3x ≤ 9
Now, divide both sides by 3:
3x/3 ≤ 9/3
x ≤ 3
So, the solution to the inequality -13 < 3x + 2 ≤ 11 is -5 < x ≤ 3.
Expressing this solution in interval notation:
(-5, 3]