# 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 $\ge$.

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. $m < n$