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3 years ago

What is the reason for each step in the solution of the inequality?

Mathematics

2 answers:

posledela3 years ago

5 0

Answer:

distributive property
combine like terms
division property of order

Step-by-step explanation:

At the second step, parentheses have been eliminated. The distributive property is used for that.

At the third step, -25-14 is replaced by -39. This process is called combine like terms.

At the last step, the inequality is divided by the coefficient of x. The division is by a negative number, so the order symbol is reversed, according to the division property of order.

Vinil7 [7]3 years ago

3 0

Answer:

Statement Reason

$4(x+6)>-5(x+5)-14$ (Given)

$4x+24>-5x-39$ (Combine like terms)

$4x+63>-5x$ (Addition property of order)

$63>-9x$ (Subtraction property of order)

$-7$ (Division property of order)

Step-by-step explanation:

The given inequality is

$4(x+6)>-5(x+5)-14$

Using distributive property, we get

$4(x)+4(6)>-5(x)-5(5)-14$

$4x+24>-5x-25-14$

Combine like terms.

$4x+24>-5x-39$

Using addition property of order, add 39 on both sides.

$4x+24+39>-5x-39+39$

$4x+63>-5x$

Using subtraction property of order, subtract 4x from both sides.

$4x+63-4x>-5x-4x$

$63>-9x$

Using division property of order, divide both sides by -9. If we multiply or divide an inequality by a negative number, then the sign of inequality will change.

$-7$

Hence proved.

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Find the solution set of the inequality: -3x + 8 < 15−3x+8<15

makkiz [27]

Answer:

x=(8/3, infinity)

Step-by-step explanation:

-3x+8<15-3x+8

-3x+8<-3x+23

8<23

15-3x+8<15

23-3x<15

-3x<-8

x>8/3

Therefore, x=(8/3, infinity)

3 0

3 years ago

Write each set of fractions in order from least to greatest. 2/3, 15/18, 1/6

SIZIF [17.4K]

Answer:

1/6, 2/3, 15/18

Step-by-step explanation:

get a common denominator

2/3 times 6

15/18 times 1

1/6 times 3

new fractions : 12/18, 15/18, 3/18

so it would be 3/18, 12/18, 15/18

then just revert it back to the original fraction and you have your answer

5 0

3 years ago

Use Lagrange multipliers to find the maximum and minimum values of (i) f(x,y)-81x^2+y^2 subject to the constraint 4x^2+y^2=9. (i

sp2606 [1]

i. The Lagrangian is

$L(x,y,\lambda)=81x^2+y^2+\lambda(4x^2+y^2-9)$

with critical points whenever

$L_x=162x+8\lambda x=0\implies2x(81+4\lambda)=0\implies x=0\text{ or }\lambda=-\dfrac{81}4$

$L_y=2y+2\lambda y=0\implies2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1$

$L_\lambda=4x^2+y^2-9=0$

If $x=0$ , then $L_\lambda=0\implies y=\pm3$ .
If $y=0$ , then $L_\lambda=0\implies x=\pm\dfrac32$ .
Either value of $\lambda$ found above requires that either $x=0$ or $y=0$ , so we get the same critical points as in the previous two cases.

We have $f(0,-3)=9$ , $f(0,3)=9$ , $f\left(-\dfrac32,0\right)=\dfrac{729}4=182.25$ , and $f\left(\dfrac32,0\right)=\dfrac{729}4$ , so $f$ has a minimum value of 9 and a maximum value of 182.25.

ii. The Lagrangian is

$L(x,y,z,\lambda)=y^2-10z+\lambda(x^2+y^2+z^2-36)$

with critical points whenever

$L_x=2\lambda x=0\implies x=0$ (because we assume $\lambda\neq0$ )

$L_y=2y+2\lambda y=0\implies 2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1$

$L_z=-10+2\lambda z=0\implies z=\dfrac5\lambda$

$L_\lambda=x^2+y^2+z^2-36=0$

If $x=y=0$ , then $L_\lambda=0\implies z=\pm6$ .
If $\lambda=-1$ , then $z=-5$ , and with $x=0$ we have $L_\lambda=0\implies y=\pm\sqrt{11}$ .

We have $f(0,0,-6)=60$ , $f(0,0,6)=-60$ , $f(0,-\sqrt{11},-5)=61$ , and $f(0,\sqrt{11},-5)=61$ . So $f$ has a maximum value of 61 and a minimum value of -60.