INEQUALITY MATERIAL, EXERCISE AND SOLUTION - mathematic.my.id

INEQUALITY MATERIAL, EXERCISE AND SOLUTION

Inequality, Exercise and Solution

Inequality Material

A. Definition

Inequation is the open mathematic sentence containing the symbol: $>$, $<$, $\le$, or $\ge$.

B. Natures Inequality

If $a>b$ then it can be taken the nature of:
1. $a+c>b+c$
2. $a-c>b-c$
3. $a.c>b.c$ for $c$ positive number
$~~~a.c < b.c$ for $c$ negative number
4. $a:c>b:c$ for $c$ positive number
$~~~a:c < b:c$ for $c$ negative number
This applies to the other symbol of inequality. Specifically for multiplication or division of negative numbers then the symbol of change direction.

C. Kinds Inequality

1. Linear Inequality

General form:
$ax+b>c$

Steps completion:
a. Less both sides with the $b$
b. Divide both sides with the $a$, then be $$x>\frac{c-b}{a}~\text{for $a$ positive number}$$ $$x<\frac{c-b}{a}~\text{for $a$ negative number}$$ This also applies to symbols $\le$ and $\ge$.

2. Quadratic Inequality

General form:
$$ax^2+bx+c>0$$ $$ax^2+bx+c<0$$
Steps completion:
1. Search the roots quadratic equality
2. If this roots $x_1$ and $x_2$ with $x_1 < x_2$ then,
3. If inequation symbol "$>$" then: $x < x_1$ or $x>x_2$
4. If inequation symbol "$<$" then: $x_1 < x < x_2$.
This also applies to symbols $\le$ and $\ge$.

3. Fractional Inequality

General form:
$$\frac{ax+b}{cx+d} > e$$
Steps completion:
a. Move $e$ to the left section, then $$\frac{ax+b}{cx+d}-e>0$$ $$\frac{(a-ce)x+(b-de)}{cx+d}>0$$ b. if $$\frac{de-b}{a-ce}>-\frac{d}{c}$$ then $x \ne -\frac{d}{c}$ and $$x < -\frac{d}{c}$$ or $$x>\frac{de-b}{a-ce}$$ c. In contrast to $$\frac{de-b}{a-ce} < -\frac{d}{c}$$


Exercise and Solution for Inequality Material

1. If $d>b$, $d>c$, and $b < c$ with $b, c, d < 0$, then ...
$$\text{A. }\frac{d}{b}>\frac{c}{d}$$ $$\text{B. } \frac{b}{c} < \frac{d}{b}$$ $$\text{C. }\frac{b}{c} > \frac{c}{d}$$ $$\text{D. } \frac{d}{b} < \frac{d}{c}$$ $\text{E. }b,~ c, \text{ and } d$ $\text{ relationship cannot be determined.}$

2. If $a < 7m < b$ and $b < 4n < c$ with $a < b < c$ then ...
A. $m=n$
B. $m < n$
C. $m$ and $n$ relationship cannot be determined
D. $m>n$
E. $m < 4n/7$

3. If $a+2 < x+p < b+2$ and $b < y+p < c$ with $a < b < c$ then ...
A. $x < y$
B. $x > y$
C. $x=y$
D. $x+y=0$
E. $x$ and $y$ relationship cannot be determined

4. If $0 < ab < 1$ and $a>0$ then the following a definite truth is ...
A. $b>1/a$
B. $a>1/b$
C. $0 < 1/a < 1/b$
D. $0 < b < 1/a$
E. C and D is true.

5. If $0 < x < 1$, the following statement that the order increase is ...
A. $\sqrt{x}$, $x$, $x^2$
B. $x^2$, $x$, $\sqrt{x}$
C. $x^2$, $\sqrt{x}$, $x$
D. $x$, $x^2$, $\sqrt{x}$
E. $x$, $\sqrt{x}$, $x^2$


6. If $3 < x < 5$ and $5 < y < 8$, then ...
A. $x>y$
B. $x < y$
C. $x=y$
D. $x$ and $y$ relationship cannot be determined
E. $x+y=8$

7. If $-2 \le x \le 7$ and $4 \le y \le 9$, then $x$ and $y$ relationship is ...
A. $x>y$
B. $x < y$
C. $x=y$
D. Cannot be determined
E. $x+y>16$

8. If $1 < a < 5$ and $1 < b < 5$ then $a$ and $b$ relationship is ...
A. $a=b$
B. $a>b$
C. $a < b$
D. Cannot be determined
E. $a \ge b$

9. If $4 < x < 8$ dan $0 < y < 1,5$ then interval $x.y$ is ...
A. $0 < x.y < 6$
B. $0 < x.y <12$
C. $1,5 < x.y < 4$
D. $1,5 < x.y < 8$
E. $1,5 < x.y < 6$

10. If $0 < x \le 5$ and $-4 \le y < 5$, then the following figures that are not included the set of value $x.y$ is ...
A. $-20 \quad ~~$ D. 25
B. $-2 \quad ~~~$ E. $-4$
C. 0


11. Jika $-5 \le p \le 8$ dan $-1 < q < 5$, then ...
A. $0 \le p^2+q^2 \le 89$
B. $0 \le p^2+q^2 < 89$
C. $26 \le p^2+q^2 < 89$
D. $26 \le p^2+q^2 \le 89$
E. $26 < p^2+q^2 \le 89$

12. If $-8 < x < 8$ and $-4 < y < 3$ then ...
A. $-4 < x-y < 5$
B. $-4 < x-y < 12$
C. $-11 < x-y < 12$
D. $-11 < x-y < 5$
E. $0 < x-y < 12$

13. Given $7 < 3x+4 < 13$, the value $x$ is ...
A. 0 $\quad ~$ D. 3
B. 1 $\quad ~$ E. Can't be determined
C. 2,5

14. If $x$ is a positive integer from inequality $2x+1>3x-3$, then a lot of value $x$ is ...
A. Not there
B. 1
C. 2
D. 3
E. 4


15. If $m^2-6m+5 < 0$ and $$\frac{n-7}{n-5} \le 0$$ $n \ne 5$ then ...
A. $m=n$
B. $m+n=0$
C. $m$ and $n$ relationship can't be determined
D. $m>n$
E. $m < n$

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