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Here are a few recommended readings before getting started with this lesson.
at most,
less than,and
at leastthat restrict the amount of the weight?
An inequality, like an equation, is a mathematical statement that compares two quantities. An inequality contains the symbols $<,$ $>,$ $≤,$ or $≥.$ There are several ways each inequality can be phrased.
Inequality Symbol  Key Phrases 

$<$  $ ∙is less thanor equal toter∙is fewer than $

$>$  $ ∙is greater thanor equal to∙is more than $

$≤$  $ ∙is less than or equal toter∙is at most∙is no more than $

$≥$  $ ∙is greater than or equal to∙is at least∙is no less than $

With an inequality, it is possible to compare any combination of two numbers, variables, numeric expressions, or algebraic expressions.
Symbol  Example  Meaning 

$<$  $x<1$  The variable $x$ is less than $1.$ 
$≤$  $x+1≤3$  The algebraic expression $x+1$ is less than or equal to $3.$ 
$>$  $2x−5>5$  The expression $2x−5$ is greater than $5.$ 
$≥$  $x≥2x+1$  The variable $x$ is greater than or equal to the expression $2x+1.$ 
It is the weekend and Tadeo is at home, flipping through channels on the TV. He decides to make up math questions about whatever is on each channel to entertain himself. On the sports channel is a program about Olympic swimmer Michael Phelps. The program claims that Phelps has a lung capacity of $12$ liters, nearly twice that of an average adult.
If Phelps's lungs are full and he exhales at a rate of $1.2$ liters per second in a swimming race, help Tadeo write an inequality that models when Phelps will still have more than $5.1$ liters of air left in his lungs.
Let $t$ be the variable representing the number of seconds Phelps exhales air from his lungs.
more than$5.1$ liters, the inequality is strict and the symbol
$>$should be used.
The nature channel is showing a program produced by the National Association of State Forestry about shade trees. Tadeo learns that both the cottonwood and the red maple are shade trees and that the diameter of a red maple trunk grows by about $0.31$ inches per year, while the diameter of a cottonwood trunk grows by about $0.64$ inches per year.
Assume that a red maple with a trunk diameter of $3.4$ inches has been planted together with a cottonwood with a trunk diameter of $2.7$ inches. Help Tadeo write an inequality that models when the red maple will be thinner than the cottonwood.
Let $t$ be the variable representing the number of years that pass.
thinner thanthe that of the cottonwood. Therefore, the inequality will be strict and the
$<$symbol should be used.
The next channel is showing reruns of one of Tadeo's favorite TV shows, Supernatural. The show features a $1967$ Chevrolet Impala. Tadeo decides to save up for a car just like this one.
After some research online, Tadeo learns that a $1967$ Chevrolet Impala is available in his town for about $$23000.$ Currently, he has $$500$ in savings. He plans to save about $$200$ each month until he can afford the car. He wants to save at least $$25000$ in case of any price changes before he can afford it. Write an inequality that models when Tadeo will have saved at least $$25000.$
Let $m$ be the variable representing the number of months Tadeo saves money.
at least$$25000.$ This means that he will need either $$25000$ or more than $$25000.$ Since the possibility of an equality exists, the inequality will be nonstrict and the symbol
$≥$should be used.
Tadeo sees a segment for a local amusement park on the news. According to the segment, an annual pass to the park is $$129.99.$ Tadeo knows that roundtrip bus fare to the amusement park is $$10$ per trip.
Tadeo wants to use some of his savings buy a pass. He hopes to visit the amusement park regularly throughout the year with his friends. Supposing he uses no more than $$500$ of his savings, write an inequality representing the number of times Tadeo can visit the park if he also has to purchase the annual pass.Let $t$ be the variable representing the number of round trips Tadeo can take on the bus.
no more than$$500.$ Therefore, the total cost must be less than or equal to $$500.$ This means that the inequality will be nonstrict and the
$≤$symbol should be used.
Several word problems have been modeled by different inequalities throughout this lesson. However, before ending the lesson, there are some properties should be mentioned. Similar to the properties of equality, inequalities also have some properties. Note that even if these properties are closely related to the properties of equality, there are important differences.
A real number can never be less than or greater than itself.
$x≮x$ and $x≯x$
For any two real numbers $x$ and $y,$ if $x$ is less than $y,$ then $y$ cannot be less than $x.$
If $x<y,$ then $y <x.$
Alternatively, if $x$ is greater than $y,$ then $y$ cannot be greater than $x.$
If $x>y,$ then $y >x.$
Let $x,$ $y,$ and $z$ be real numbers. If $x$ is less than $y$ and $y$ is less than $z,$ then $x$ is less than $z.$
If $x<y$ and $y<z,$ then $x<z.$
This property also applies to other types of inequalities — $>,$ $≤,$ and $≥.$