Solve each equation.
1. (|3x -4|)/(-5) = 6
A: no solution
2. (|2x -5|) = x + 3
A: x = 8 or x = 2/3
Solve each inequality.
3. |5x + 15| > 20
A: x > 1 and x < -7
4. [(|x - 2|)/ (4) greater than or equal to 5
A: -18 < x < 22
5. (-3) x (|5x - 8|)- 5 greater than or equal to 6
A: no solution?
6. |12 - 4x| - 4 > 20
A: x < -3 and x > 9
1. |3x-4}/(
2. |2x-5| = x + 3
2x-5 = x+3
X = 8.
-2x+5 = x+3
-3x = -2
X = 2/3
3. |5x+15| > 20
5x+15 > 20
5x > 20-15
X > 1
-5x-15 = > 20
-5x > 20 + 15
X < -7
Or 1 < X < -7
4. |x-2|/4 => 5
(x-2)/4 => 5
x-2 => 20
X => 22.
(-x+2)/4 => 5
-x + 2 => 20
-x => 18
X =< -18
5. (-3)*|5x-8}-5 => 6
(-3)*|5x-8| => 11
|5x-8| =< 11/-3 ? No solution.
The absolute value of a number is always positive.
6. |12-4x|-4 > 20
|12-4x| > 24
12-4x > 24
-4x > 12
X < -3
-12+4x > 20
4x > 32
X > 8 Does not satisfy the inequality.
1. |3x-4|/(-5) = 6
(3x-4)/(-5) = 6
3x-4 = -30
3x = -26
X = -26/3 Does not satisfy Eq.
(-3x+4)/(-5) = 6
-3x+4 = -30
-3x = -34
X = -34/-3 = 34/3 Does not satisfy Eq.
To solve each equation or inequality involving absolute value, we need to isolate the absolute value expression and then split it into two separate cases - one where the expression within the absolute value is positive, and one where it is negative. We'll go through each problem step by step:
1. (|3x - 4|)/(-5) = 6
To isolate the absolute value expression, multiply both sides of the equation by -5:
|3x - 4| = -30
Since an absolute value cannot be negative, there are no values of x that satisfy this equation. Therefore, there is no solution.
2. (|2x - 5|) = x + 3
To isolate the absolute value expression, we don't need to make any changes. Now we split it into two cases:
Case 1: 2x - 5 is positive
|2x - 5| = 2x - 5
Substituting this back into the original equation:
2x - 5 = x + 3
Solving for x:
x = 8
Case 2: 2x - 5 is negative
|2x - 5| = -(2x - 5) = -2x + 5
Substituting this back into the original equation:
-2x + 5 = x + 3
Solving for x:
3x = 2
x = 2/3
So the solutions are x = 8 and x = 2/3.
Now let's move on to the inequalities:
3. |5x + 15| > 20
To isolate the absolute value expression, we separate it into two cases:
Case 1: 5x + 15 is positive
|5x + 15| = 5x + 15
Substituting this back into the original inequality:
5x + 15 > 20
Solving for x:
x > 1
Case 2: 5x + 15 is negative
|5x + 15| = -(5x + 15) = -5x - 15
Substituting this back into the original inequality:
-5x - 15 > 20
Solving for x:
x < -7
So the solutions are x > 1 and x < -7.
4. |x - 2|/4 ≥ 5
To isolate the absolute value expression, we multiply both sides of the inequality by 4:
|x - 2| ≥ 20
Now we split it into two cases:
Case 1: x - 2 is positive
|x - 2| = x - 2
Substituting this back into the original inequality:
x - 2 ≥ 20
Solving for x:
x ≥ 22
Case 2: x - 2 is negative
|x - 2| = -(x - 2) = -x + 2
Substituting this back into the original inequality:
-x + 2 ≥ 20
Solving for x:
x ≤ -18
So the solutions are -18 < x < 22.
5. (-3) × |5x - 8| - 5 ≥ 6
To isolate the absolute value expression, we add 5 to both sides of the inequality:
(-3) × |5x - 8| ≥ 11
Since the left side is multiplied by -3, the absolute value expression cannot be negative. Therefore, there is no solution.
6. |12 - 4x| - 4 > 20
To isolate the absolute value expression, we add 4 to both sides of the inequality:
|12 - 4x| > 24
Now we split it into two cases:
Case 1: 12 - 4x is positive
|12 - 4x| = 12 - 4x
Substituting this back into the original inequality:
12 - 4x > 24
Solving for x:
x < -3
Case 2: 12 - 4x is negative
|12 - 4x| = -(12 - 4x) = -12 + 4x
Substituting this back into the original inequality:
-12 + 4x > 24
Solving for x:
x > 9
So the solutions are x < -3 and x > 9.