1) To solve the pair of inequalities:
5/2 - x/2 <= 3
1/2x + 2 > x - 1
We can simplify the first inequality by multiplying both sides by 2:
5 - x <= 6
-x <= 1
x >= -1
For the second inequality, we can start by subtracting 1/2x from both sides:
1/2x + 2 - 1/2x > -1
Simplifying, we get:
2 > -1
This is always true, so there are no restrictions on x from this inequality. Therefore, the solution set for both inequalities combined is:
x >= -1
2) The unshaded region R can be described by the inequalities:
x > 0
y > 0
y < 2 - x
y > x/2
The first two inequalities ensure that the region is in the first quadrant. The third inequality defines the upper boundary of the region as a line with slope -1 passing through (2,0). The fourth inequality defines the lower boundary of the region as a line with slope 1/2 passing through (0,0).
3) Let x be the number of cows and y be the number of goats. Then the constraints are:
x + y <= 3
(12000x + 4000y)/3 <= 240000
x <= 2y
x > 5
The first constraint ensures that the farmer doesn't buy more than 3 animals. The second constraint limits the total cost to no more than sh 240000. The third constraint ensures that the number of cows is at most twice the number of goats. The fourth constraint requires the farmer to buy more than 5 cows.
4) To simplify the left-hand side of the equation:
x^h/y^(2/3) ÷ y^k/yx^3
= x^(h-3)/y^(2/3-k)
Setting this equal to x^3y^-4, we get:
h-3 = 3
2/3 - k = -4
Solving for h and k, we get:
h = 6
k = 10/3
So, h+k = 6 + 10/3 = 28/3.
5) Taking the logarithm (base 2) of both sides of the first equation:
x*log2(8) = (2y+1)*log2(4)
x = (2y+1)/3
Taking the logarithm (base 3) of both sides of the second equation:
2x*log3(27) = (y-3)*log3(9)
2x*3 = y-3
x = (y-3)/6
Setting these two expressions for x equal to each other:
(y-3)/6 = (2y+1)/9
9(y-3) = 6(2y+1)
9y - 27 = 12y + 6
-3y = 33
y = -11
Substituting into either equation to solve for x:
8^x = 4^(2(-11)+1) = 4^(-21)
x*log2(8) = -21*log2(4)
x = -21/3 = -7
So, the solution is (x,y) = (-7,-11).
6) We can simplify the expression using exponent laws:
8^(2/3) + 4^(3/2) ÷ 16^(-3/4)
= (2^3)^(2/3) + (2^2)^(3/2) ÷ (2^4)^(-3/4)
= 2^2 + 2^3 ÷ 2^(-3)
= 4 + 8/8
= 5
7) We can simplify the expression using exponent laws:
(16/81)^(-3/4) × 512^(2/3)
= (2^4/3^4)^(-3/4) × (2^9)^(2/3)
= 2^-3 × 2^6
= 2^3
So, the answer is 2^3.
8) We can rewrite the equation as:
3^(1-2x) - 10(3^x) + 3 = 0
Letting u = 3^x, we can substitute to get:
u^2 - 10u + 3 = 0
Solving this quadratic equation for u, we get:
u = (10 ± √88)/2
u = 5 ± √22
Substituting back and solving for x, we get:
3^x = 5 + √22 or 3^x = 5 - √22
Taking the logarithm (base 3) of both sides, we get:
x = log3(5 + √22) or x = log3(5 - √22)
These are the exact solutions.
9) From the first equation, we can rewrite as:
2^(3x) = 2^(4y+2)
So, 3x = 4y+2, or x = (4y+2)/3.
From the second equation, we can rewrite as:
(3^2)^(2x) = (3^2)^(-3)
So, 2x = -3/2, or x = -3/4.
Substituting into the expression for x from the first equation:
(4y+2)/3 = -3/4
Solving for y, we get:
y = -11/8
So, the solution is (x,y) = (-3/4, -11/8).
1)What integral value of x satisfy the following pairs of inequalities ;5/2-x/2<\=3
1/2x+2>x-1
2)Write down four inequalities which fully describe the unshaded region R in the figure S below.
3)A farmer wishes to purchase some goats and cows.He can buy at most 3 either animals.On average a cow and a goat cost sh 12000 and sh 4000 respectively.He has sh 240000 to spend and the number of cows should be at most twice the number of goats.He must buy more than five cows.List all the inequalities to represent the above information.
4)x^(h)/y^(2/3)÷y^(k)/yx^(3)=x^(3)y^(-4),find h+k
5) Solve 8^x=4^(2y+1) and 27^(2x)=9^(y-3), giving your answer as an exact fraction.
6) Without using mathematical table or calculator evaluate; 8^(2/3)+4^(3/2)÷16^(-3/4)
7) without using mathematical table or calculator evaluate, leaving your answer in power form;(16/81)^(-3/4)×512^(2/3)
8)Solve for x in 3^(-2x+1)-10(3^x)+3=0, without using tables
9)solve for 8^(x)=4^(2y+1) and 27^(2x)=9^(-3) giving your answer as an exact fraction
1 answer