Solve for x. The solutions in interval notation are of the form (a, b). State a. (Enter an exact number.)
abs(3 x + 15) < 30
|3x+15| < 30
case 1:
3x+15 < 30
3x < 15
x < 5
case 2:
-(3x+15) < 30
3x+15 > -30
3x > -45
x > -15
so, the solution set is (-15,5), as shown here:
http://www.wolframalpha.com/input/?i=|3x%2B15|+%3C+30
To solve the absolute value inequality "abs(3x + 15) < 30", we can start by removing the absolute value by splitting it into two separate inequalities.
For the given inequality "abs(3x + 15) < 30", we can write it as two separate inequalities:
1) 3x + 15 < 30
2) -(3x + 15) < 30
Now let's solve each inequality step-by-step:
1) 3x + 15 < 30
Subtract 15 from both sides:
3x + 15 - 15 < 30 - 15
3x < 15
Divide both sides by 3:
(3x) / 3 < 15 / 3
x < 5
So, the first inequality is x < 5.
2) -(3x + 15) < 30
Distribute the negative sign:
-3x - 15 < 30
Add 15 to both sides:
-3x - 15 + 15 < 30 + 15
-3x < 45
Divide both sides by -3 (Note: When dividing by a negative number, the inequality sign will flip):
(-3x) / -3 > 45 / -3
x > -15
So, the second inequality is x > -15.
Therefore, the solution to the absolute value inequality "abs(3x + 15) < 30" is given by the intersection of the solutions to the two separate inequalities:
x < 5 and x > -15
In interval notation, this can be written as (-15, 5). Therefore, the answer is a = -15.
To solve the inequality abs(3x + 15) < 30, we need to isolate the absolute value expression and remove the absolute value bars. Here's how we can do it step by step:
1. Start by writing the absolute value as two separate inequalities, one with a positive sign and one with a negative sign:
3x + 15 < 30 and -(3x + 15) < 30
2. Solve each inequality separately to find the two sets of solutions:
Solving the first inequality:
3x + 15 < 30
Subtract 15 from both sides:
3x < 15
Divide both sides by 3 (since the coefficient of x is 3 and we want to solve for x):
x < 5
Solving the second inequality:
-(3x + 15) < 30
Distribute the negative sign to both terms inside the parentheses:
-3x - 15 < 30
Add 15 to both sides:
-3x < 45
Divide both sides by -3. Remember, when we divide or multiply both sides of an inequality by a negative number, we must reverse the inequality sign:
x > -15
Therefore, the solutions to the inequality are: -15 < x < 5.
However, notice that the problem asks for the solutions in interval notation of the form (a, b), where a and b are exact numbers. In this case, we don't have any strict inequalities (< or >), so we need to adjust the endpoints of our interval. To do this, we use square brackets [] instead of parentheses () when the inequality includes equality (≤ or ≥). Since the inequality does not include equality, we need to use strict inequality symbols (< and >) in interval notation.
So, the solution in interval notation would be: (-15, 5). The exact number a is -15.