Using interval notation to express all real numbers less than or equal to
a Or greater than or equal to
b
Write the interval expressing all real numbers less than or equal to
$\text{\hspace{0.17em}}\mathrm{-1}\text{\hspace{0.17em}}$ or greater than or equal to
$\text{\hspace{0.17em}}1.$
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at
$\text{\hspace{0.17em}}-\infty \text{\hspace{0.17em}}$ and ends at
$\text{\hspace{0.17em}}\mathrm{-1},$ which is written as
$\text{\hspace{0.17em}}\left(-\infty ,\mathrm{-1}\right].$
The second interval must show all real numbers greater than or equal to
$\text{\hspace{0.17em}}1,$ which is written as
$\text{\hspace{0.17em}}\left[1,\infty \right).\text{\hspace{0.17em}}$ However, we want to combine these two sets. We accomplish this by inserting the union symbol,
$\cup ,$ between the two intervals.
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the
addition property and the
multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
These properties also apply to
$\text{\hspace{0.17em}}a\le b,$$a>b,$ and
$\text{\hspace{0.17em}}a\ge b.$
Demonstrating the addition property
Illustrate the addition property for inequalities by solving each of the following:
(a)
$x-15<4$
(b)
$6\ge x-1$
(c)
$x+7>9$
The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
Solving inequalities in one variable algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.
Solving an inequality algebraically
Solve the inequality:
$\text{\hspace{0.17em}}13-7x\ge 10x-4.$
Solving this inequality is similar to solving an equation up until the last step.
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387