Solve the inequality. Write the solution in interval notation.
|2 - 3x| <= 5
Select the correct choice below, and fill in the answer box if necessary.
OA. The solution set is ___ (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
B. There is no solution.
To solve the inequality |2 - 3x| <= 5, we can break it down into two separate inequalities:
1) 2 - 3x <= 5
2) -(2 - 3x) <= 5
For inequality 1, we subtract 2 from both sides:
-3x <= 3
Dividing both sides by -3 (and reversing the inequality since we are dividing by a negative number) gives:
x >= -1
For inequality 2, we distribute the negative sign and simplify:
-2 + 3x <= 5
Adding 2 to both sides gives:
3x <= 7
Dividing both sides by 3 gives:
x <= 7/3
Putting the solution to both inequalities together, we have:
x >= -1 and x <= 7/3
In interval notation, this can be written as:
[-1, 7/3]
To solve the inequality |2 - 3x| <= 5, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: 2 - 3x >= 0
In this case, we have:
2 - 3x <= 5
-3x <= 3
x >= -1
Case 2: 2 - 3x < 0
In this case, we have:
-(2 - 3x) <= 5
-2 + 3x <= 5
3x <= 7
x <= 7/3
So, the solution set is {-1} U (-∞, 7/3] in interval notation.
Therefore, the correct choice is OA. The solution set is (-∞, 7/3]
To solve the inequality |2 - 3x| <= 5, we need to break the problem down into two cases:
Case 1: 2 - 3x >= 0
In this case, the absolute value of 2 - 3x is equal to 2 - 3x. So the inequality becomes:
2 - 3x <= 5
Solving this inequality, we subtract 2 from both sides:
-3x <= 3
Then, divide both sides by -3. Since we are dividing by a negative number, the inequality sign flips:
x >= -1
So, for Case 1, the solution is x >= -1.
Case 2: 2 - 3x < 0
In this case, the absolute value of 2 - 3x is equal to -(2 - 3x). So the inequality becomes:
-(2 - 3x) <= 5
To remove the negative sign, we need to multiply both sides by -1. Remember, when you multiply an inequality by a negative number, the inequality sign flips:
2 - 3x >= -5
Next, subtract 2 from both sides:
-3x >= -7
Finally, divide both sides by -3. Since we are dividing by a negative number, the inequality sign flips again:
x <= 7/3
So, for Case 2, the solution is x <= 7/3.
To find the overall solution, we need to consider both cases. The solution set is the intersection of the solution sets for each case, which means we take the common values. Therefore, the solution is:
x >= -1 and x <= 7/3
By combining these inequalities, we can write the solution in interval notation as [-1, 7/3].
Therefore, the correct choice is OA. The solution set is [-1, 7/3].