-7≤ 2x + 7 ≤ 31
To solve this inequality, we need to isolate the variable, x.
First, let's start by subtracting 7 from all parts of the inequality to get:
-7 - 7 ≤ 2x + 7 - 7 ≤ 31 - 7
-14 ≤ 2x ≤ 24
Next, let's divide all parts of the inequality by 2 to get:
-14/2 ≤ 2x/2 ≤ 24/2
-7 ≤ x ≤ 12
Therefore, the solution to the inequality is -7 ≤ x ≤ 12.
3x + 10 < -2 or 13 < 3x + 10
To solve this compound inequality, we will solve each inequality separately and then combine their solutions.
First, let's solve the inequality 3x + 10 < -2:
Subtracting 10 from both sides, we get:
3x + 10 - 10 < -2 - 10
3x < -12
Next, dividing both sides by 3, we get:
(1/3)3x < (-12/3)
x < -4
Now, let's solve the inequality 13 < 3x + 10:
Subtracting 10 from both sides, we get:
13 - 10 < 3x + 10 - 10
3 < 3x
Dividing both sides by 3, we get:
(1/3)3 < (1/3)3x
1 < x
Finally, combining the solutions, we have:
x < -4 or 1 < x
So, the solution to the compound inequality is x < -4 or 1 < x.
-x -3 ≥ -2 or -x -3 ≤ -9
Let's solve each inequality separately:
Inequality 1: -x - 3 ≥ -2
Adding 3 to both sides, we get:
-x - 3 + 3 ≥ -2 + 3
-x ≥ 1
Multiplying both sides by -1 (which reverses the inequality because we are multiplying by a negative number), we get:
(-1)(-x) ≤ (-1)(1)
x ≤ -1
Inequality 2: -x - 3 ≤ -9
Adding 3 to both sides, we get:
-x - 3 + 3 ≤ -9 + 3
-x ≤ -6
Multiplying both sides by -1 (which reverses the inequality), we get:
(-1)(-x) ≥ (-1)(-6)
x ≥ 6
Combining the solutions from both inequalities, we have:
x ≤ -1 or x ≥ 6
So, the solution to the compound inequality is x ≤ -1 or x ≥ 6.
-3x - 5 ≥-2 or -23 ≥ -3x -5
Let's solve each inequality separately:
Inequality 1: -3x - 5 ≥ -2
Adding 5 to both sides, we get:
-3x - 5 + 5 ≥ -2 + 5
-3x ≥ 3
Now, dividing both sides by -3 (which reverses the inequality because we are dividing by a negative number), we get:
(-1/3)(-3x) ≤ (1/3)(3)
x ≤ -1
Inequality 2: -23 ≥ -3x - 5
Adding 5 to both sides, we get:
-23 + 5 ≥ -3x - 5 + 5
-18 ≥ -3x
Dividing both sides by -3 (which reverses the inequality), we get:
(-1/3)(-18) ≤ (-1/3)(-3x)
6 ≤ x
Combining the solutions from both inequalities, we have:
x ≤ -1 or 6 ≤ x
So, the solution to the compound inequality is x ≤ -1 or 6 ≤ x.
-45 < -5x -5 or -65 ≥ -5x -5
Let's solve each inequality separately:
Inequality 1: -45 < -5x - 5
Adding 5 to both sides, we get:
-45 + 5 < -5x - 5 + 5
-40 < -5x
Dividing both sides by -5 (which reverses the inequality because we are dividing by a negative number), we get:
(-1/5)(-40) > (-1/5)(-5x)
8 > x
Inequality 2: -65 ≥ -5x - 5
Adding 5 to both sides, we get:
-65 + 5 ≥ -5x - 5 + 5
-60 ≥ -5x
Dividing both sides by -5 (which reverses the inequality), we get:
(-1/5)(-60) ≤ (-1/5)(-5x)
12 ≤ x
Combining the solutions from both inequalities, we have:
x > 8 or x ≥ 12
So, the solution to the compound inequality is x > 8 or x ≥ 12.