From Equations to Inequalities

From Equations to Inequalities exercise

• Compare the amount of fishes skeeter fished with the catching limit of $999$ lbs. per month.

  Hints

  Let's look at the following examples:

  • $12>10$, where $>$ is the greater than sign.
  • $10<12$, where $<$ is the less than sign.

  You can compare an equation with a scale: for equality, on both sides of the equals sign we must have the same value. So a scale would be in balance.

  Solution

  The fisherman Skeeter has to take care of a very important regulation: the amount of fishes he is able to catch is limited to $999$ lbs. a month.

  In the first month Skeeter caught exactly $999$ lbs. That's the same as the given limit. So we use the equals sign:

  $999=999$.

  The value on the left side is equal to the one on the right side.

  The second month he caught $800$ lbs. of fishes. This is less than the allowed amount. Thus we use the less than sign:

  $800<999$.

  The value on the left side is smaller than the one on the right side.

  In the third month Skeeter caught $1250$ lbs of fish. Oh, that's illegal. It's more than the allowed amount. Here we use the greater than sign:

  $1250>999$.

  The value on the left side is greater than the one on the right side.

• Express the given problem exactly.

  Hints

  The less than sign $<$ indicates that the value on the left side of the sign is less than the one on the right side.

  For example, $10<12$.

  But $=$ is not included.

  If we include $=$ with $<$ sign, we write it as $\le$.

  So, for example, we have that $12\le 12$.

  Solution

  Please help Skeeter to find out the correct expression which represents the allowed amount of lbs. of fishes which can be caught per month.

  He already knows that $800$ lbs. is OK, and that $999$ lbs. is allowed too.

  That means that the regulation is fulfilled for amounts less than $999$ lbs. and also for $999$ lbs. itself.

  We can express these regulations using the less than or equal to sign $\le$.

  If $f$ the amount of lbs. of fishes caught by Skeeter in a month, then if $f$ is less than or equal to $999$ the regulations are followed:

  $f\le 999$.

• Find the mistakes.

  Hints

  Keep the following in mind:

  If you use the equals sign, $=$, the numbers on both sides of the sign must be the same.

  Let's have a look at the following examples:

  • $23=23$
  • $23\le 23$
  • $23\ge 23$
  • $22 < 23$
  • $22\le 23$
  • $22>21$
  • $22\ge 21$

  The following sentences aren't true:

  • $23>23$
  • $22<21$
  • $21>22$
  • $21=22$

  Keep the following in mind: if two numbers are equal, you can also use $\le$ as well as $\ge$ because both include the equals sign.

  Solution

  The relation signs express if two numbers are equal, or if one is greater or less than the each other.

  Here you see some examples for the equal sign:

  • $234=234$
  • $123=123$
  • $345=345$

  Each time you can also use $\le$ or $\ge$ because both signs include the equals sign:

  • $234\le234$ as well as $234\ge234$
  • $123\le123$ as well as $123\ge123$
  • $345\le345$ as well as $345\ge345$

  And we also have that $678=678$, as a number is not greater or less than itself.

  Last we still have,

  • $789<890$ or $789\le 890$
  • $890>123$ or $890\ge 123$

• Find the right expression.

  Hints

  Pay attention to if equality is allowed or not.

  Let's have a look at an example: if Paul is only allowed to eat $4$ tacos then he can eat no tacos at all, or just one, two, three, or four tacos. We can write this as follows, using $t$ for the number of tacos: $t\le 4$.

  Solution

  Speed limit:

  A race of $60$ mph or less is allowed. So we can use $\le$ with the variable $s$ for the speed: $s\le 60$.

  Alcohol:

  From $21$ years or older means that alcohol is allowed to be bought by those who are $21$ years or older than $21$. Mathematically we can write it like this as $y\ge 21$, where $y$ is the number of years.

  Duty free:

  Here we have an upper bound of $200$ dollars. Let $a$ be the variable for the amount. Then we have that $a\le 200$.

• Recall the definitions of the mathematical relation signs.

  Hints

  $999=999$

  This is an example for the $=$ sign.

  • $800<999$
  • $999\not < 999$

  When we want both $<$ and $=$ to be true, we use $\le$:

  • $800\le 999$
  • $999\le 999$

  • $1250>999$
  • $999\not > 999$

  When we want both $>$ and $=$ to be true, we use $\ge$:

  • $1250\ge 999$
  • $999\ge 999$

  Solution

  Here you see the summary for all relation signs:

  $=$ stands for is equal to. For example, $999=999$.

  $<$ stands for is less than. For example, $800<999$.

  $\le$ stands for is less than or equal to. For example, $800\le 999$ as well as $999\le 999$.

  $>$ stands for is greater than. For example, $1250>999$.

  $\ge$ stands for is greater than or equal to. For example, $1250\ge 999$ and also $999\ge 999$.

• Explain the expression.

  Hints

  Pay attention to $8<x$; $8$ is not included.

  $8<x\le9$ stands for

  • $8<x$ and
  • $x\le 9$.

  Both inequalities must be fulfilled.

  For example, $10$ is greater than $8$, but $10$ isn't less than or equal to $9$.

  So $10$ am isn't a possible arriving time.

  All times less than or equal $8$ am aren't possible arriving times.

  All times greater than $9$ aren't possible arriving times.

  Solution

  Poor Mia. Skeeter is so enthusiastic about inequalities. Thus, she get's the time she should arrive as an inequality. Let $x$ be the time she arrives:

  $8<x\le 9$.

  What does this mean?

  Let's start with the left side: $8$ am.

  • Mia should not arrive before $8$ am.
  • She also should not arrive at $8$ am because $8$ am is not included. You can recognize this by the $<$ sign.

  Next we look at the left side:

  • Mia should not arrive later than $9$ am.
  • $9$ am itself is a possible arriving time.

  Last we can consider any time between $8$ am and $9$ am, so

  • $8:01$ am
  • $8:02$ am
  • ...
  • $8:30$ am
  • ...
  • $8:59$ am and
  • $9:00$ am

  are possible arriving times.