Solve each of the following equations. Show its solution set on a number line.
1-I3p+1I=0
Thank you!
To solve this equation, we need to isolate the absolute value term |3p-1| and then consider its two possible cases:
1-|3p-1| = 0
or
-1+|3p-1| = 0
To isolate the absolute value term, we can rewrite the equation as:
|3p-1| = 1
Now we consider the two cases:
Case 1:
3p-1 = 1
or
3p-1 = -1
Solving for p in each case, we get:
p = 2/3
or
p = 0
Case 2:
-3p+1 = 1
or
-3p+1 = -1
Solving for p in each case, we get:
p = 0
or
p = 2/3
Therefore, the solution set for the equation 1-|3p-1| = 0 is {0, 2/3}.
To show the solution set on a number line, we can mark 0 and 2/3 on the line and shade the corresponding intervals:
<=======o-----------o=======>
-1 0 2/3 1
To solve the equation 1 - |3p + 1| = 0, we need to consider two cases:
1) When 3p + 1 ≥ 0:
In this case, the absolute value |3p + 1| is positive or zero. So the equation becomes:
1 - (3p + 1) = 0
-3p = 0
p = 0
2) When 3p + 1 < 0:
In this case, the absolute value |3p + 1| becomes negative. To solve for negative absolute value, we have to change the sign and solve the equation without the absolute value. So the equation becomes:
1 - (-(3p + 1)) = 0
1 + 3p + 1 = 0
3p + 2 = 0
3p = -2
p = -2/3
The solution set for the equation 1 - |3p + 1| = 0 is {0, -2/3}.
Now let's represent these solutions on a number line:
-∞ ---------------●--●--------------- +∞
-2/3 0
The solution set {0, -2/3} is marked on the number line. The open circles represent that the values are not included in the solution set because the equation is equal to zero, not greater than or equal to zero.
To solve the equation 1 - |3p + 1| = 0, we need to isolate the absolute value expression and then solve for p.
Step 1: Remove the absolute value by writing two separate equations, one for when the expression inside the absolute value is positive and one for when it is negative.
When 3p + 1 is positive:
1 - (3p + 1) = 0
-3p - 1 = 0
-3p = 1
p = -1/3
When 3p + 1 is negative:
1 - (-3p - 1) = 0
1 + 3p + 1 = 0
3p + 2 = 0
3p = -2
p = -2/3
So we have two solutions: p = -1/3 and p = -2/3.
Step 2: Represent the solutions on a number line.
-2/3 -1/3
--------------
-----------------o-------------------o-------------------
The open circles represent the solutions, -2/3 and -1/3. You can shade the area between them to show the solution set.