Help inequalities

• Nov 29th 2006, 02:32 AM
jerryramos
Help inequalities

7x-5<4

4/x -4>5
• Nov 29th 2006, 03:53 AM
Soroban
Hello, Jerry!

The general rule for solving inequalities is something like:

. . Solve as if it were an equation except: when multiplying or
. . . . dividing by a negative quantity, reverse the inequality.

Quote:

$\displaystyle 7x - 5\:\leq \:4$

We are given: .. . $\displaystyle 7x - 5 \;\leq \;4$

Add 5 to both sides: . $\displaystyle 7x\;\leq\;9$

Divide by 7:. . . . . . . .$\displaystyle \boxed{x \;\leq \;\frac{9}{7}}$

Quote:

$\displaystyle \frac{4}{x} -4\:\geq\:5$

There is one more rule:

. . When taking reciprocals of both sides of an inequality,
. . . . reverse the inequality.

For example: .$\displaystyle 2 < 4\quad\Rightarrow\quad \frac{1}{2} > \frac{1}{4}$

We are given: . . . $\displaystyle \frac{4}{x} - 4 \;\geq \;5$

Add 4 to both sides: . $\displaystyle \frac{4}{x}\;\geq\;9$

Take reciprocals: . . . $\displaystyle \frac{x}{4} \;\leq \;\frac{1}{9}$ .
(reverse the sign)

Multiply by 4: . . . . . .$\displaystyle \boxed{x\;\leq\;\frac{4}{9}}$