1. 8.1(3.14)
2. Evaluate 3t(t + 2) - 3t^2 for t = 19
3. 3(x + 1) = 9 + 2x
Solve each inequality. Graph the solution.
1. 2(5 - 3x) < x - 4(3 - x)
2. 10 > 7 + 3x and 9 - 4x < 1
3. 3 > 2x or x - 4 > 2
1. 8.1(3.14) = 25.134
2. Evaluate 3t(t + 2) - 3t^2 for t = 19:
3(19)(19 + 2) - 3(19^2)
= 3(19)(21) - 3(361)
= 1197 - 1083
= 114
3. 3(x + 1) = 9 + 2x:
3x + 3 = 9 + 2x
x + 3 = 9
x = 6
Solving inequalities:
1. 2(5 - 3x) < x - 4(3 - x):
10 - 6x < x - 12 + 4x
-5x < -2x - 2
-3x < -2
x > 2/3
2. 10 > 7 + 3x and 9 - 4x < 1:
3x < 3
x < 1
9 - 4x < 1
-4x < -8
x > 2
Solutions: x < 1 and x > 2
3. 3 > 2x or x - 4 > 2:
2x < 3
x < 3/2
x - 4 > 2
x > 6
Solution: x < 3/2 or x > 6
Name the subset(s) of real numbers to which each number belongs.
12 7/8
12 belongs to the subset of real numbers called integers.
7/8 belongs to the subset of real numbers called rational numbers.
8.1(3.14)
Name the subset(s) of real numbers to which each number belongs.
8.1(3.14) is a multiplication of two numbers, 8.1 and 3.14.
Both 8.1 and 3.14 belong to the subset of real numbers called irrational numbers, as they cannot be expressed as a fraction and have an infinite number of decimal places.
Solve each equation. Check for extraneous solutions.
3|x + 10| = 6
Solve each inequality. Graph the solution.
4|y - 9|>36
1/2|x + 2| > 6
To solve the equation 3|x + 10| = 6, we can split it into two cases based on the absolute value expression:
Case 1: x + 10 is non-negative (x + 10 ≥ 0)
In this case, the equation becomes 3(x + 10) = 6:
3x + 30 = 6
3x = -24
x = -8
However, we need to check if x = -8 satisfies the absolute value inequality. Plugging it back into the equation:
3|-8 + 10| = 6
3|2| = 6
6 = 6
Since -8 satisfies the equation, it is a valid solution.
Case 2: x + 10 is negative (x + 10 < 0)
In this case, the equation becomes 3(-x - 10) = 6 (we multiply by -1 to remove the negative sign):
-3x - 30 = 6
-3x = 36
x = -12
Checking if x = -12 satisfies the absolute value inequality:
3|-12 + 10| = 6
3|-2| = 6
6 = 6
Thus, the solution to the equation 3|x + 10| = 6 is x = -8 and x = -12.
To solve the inequality 4|y - 9| > 36, we can similarly split it into two cases based on the absolute value expression:
Case 1: y - 9 is non-negative (y - 9 ≥ 0)
In this case, the inequality becomes 4(y - 9) > 36:
4y - 36 > 36
4y > 72
y > 18
Case 2: y - 9 is negative (y - 9 < 0)
In this case, the inequality becomes 4(-y + 9) > 36:
-4y + 36 > 36
-4y > 0
y < 0
Thus, the solution to the inequality 4|y - 9| > 36 is y < 0 or y > 18.
To solve the inequality 1/2|x + 2| > 6, we can again split it into two cases based on the absolute value expression:
Case 1: x + 2 is non-negative (x + 2 ≥ 0)
In this case, the inequality becomes 1/2(x + 2) > 6:
1/2x + 1 > 6
1/2x > 5
x > 10
Case 2: x + 2 is negative (x + 2 < 0)
In this case, the inequality becomes 1/2(-x - 2) > 6:
-1/2x - 1 > 6
-1/2x > 7
x < -14
Thus, the solution to the inequality 1/2|x + 2| > 6 is x < -14 or x > 10.