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Here are a few recommended readings before getting started with this lesson.
Kevin loves to go fishing with his father. They usually fish on a nearby lake. At the lake, small fishing boats are rented to groups of tourists for daily fishing trips.
For safety reasons, each boat can carry at most $600$ pounds of weight. Additionally, each boat can hold a maximum of five people.
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 $2x+1.$ 
This lesson will focus on inequalities of the following forms, where $a$ is a number.
Kevin and his father manage to find a boat so that the two of them can go fishing.
at least?
more than?
is at least.This phrase represents the inequality symbol $≥.$ Then, the inequality can be written as follows.
longer than.This phrase can also be interpreted as
more than.It follows that the inequality symbol to use is $>.$ Then, the inequality could be written as follows.
The applet shows an inequality of the form $x<a,$ $x>a,$ $x≤a,$ or $x≥a.$ Determine if the given number is a solution to the inequality shown.
$x$ is greater than or equal to $6.$It is a nonstrict inequality, so $x=6$ is a solution. There are two possible cases when representing a number on a number line.
For the given inequality, a closed circle $(∙)$ is placed at $6$ because it is a solution.
Every value of $x$ greater than $6$ has to be included in the graph. Since greater numbers lie to the right on the number line, this is graphed as an arrow pointing to the right.
This is the graph of the inequality. Examine the graphs of solutions sets of different inequalities.Kevin caught as many fish as he wanted in less than $5$ hours.
is at least?
is less than,is represented by the symbol $<.$ This means that the inequality can be written as follows.
$t$ is less than $5.$This is a strict inequality, so $x=5$ is not a solution. Since $5$ is not a solution, an open circle $∘$ is used at that point.
Every value of $t$ less than $5$ has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.
This is the graph of the inequality.
After sitting in the boat for so long, Kevin wants to go for a walk by the lake. He is curious about how far he and his father can throw stones.
$k$ is less than or equal to $17.$This can be expressed algebraically as follows.
$f$ is less than $35.$This can be expressed algebraically as follows.
Examine the given graph and determine its inequality.
Similar to equations, inequalities are mathematical expressions. Inequalities are useful for modeling a constraint or condition in a realworld problem. Consider the situation presented at the beginning of the lesson. Boats are rented to groups of tourists on the lake where Kevin and his father go fishing.
Two facts are known about the boats. Each boat can carry up to $600$ pounds and hold up to five people.
up to.Is the given number a solution to the inequality?
Every value of $w$ less than $600$ has to be included in the solution set on the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.
This is the graph of the inequality.
Every value of $p$ less than $5$ has to be included in the solution set. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.
In the examples solved throughout this lesson, the variables are considered as real numbers and conditions other than those specified in the examples are ignored. For example, a negative number of people or any partial number of people would not make sense in Part B of this exercise. These constraints can be added to the solution set.Now go through the examples again and determine in which examples negative values are meaningless or only integers make sense for the solution sets.