Inequality Symbols: <, >, ≤, ≥

Inequality Symbols: <, >, ≤, ≥ exercise

• Explain the inequality symbols.

  Hints

  Here you see a number line for $>65$.

  $\ge~$ is the same as $~>~$ including the $~=~$ relation.

  Here you see a number line for $\le 55$.

  Pay attention to the circle.

  Solution

  To distinguish the inequality symbols:

  • $<~$ for the less than relation. You see the regarding number line beside. The fact, that Christopher spends less than 15 days on his trip, is represented by an empty circle.
  • $\le$ for the less than or equal relation. The difference between this symbol and the $<$-symbol is the $=$ sign. This can be seen by the filled circle.
  • $>$ for the greater than relation. Similar to the $<$ relation the corresponding circle is empty.
  • $\ge$ for the greater than or equal relation. It also includes the $=$ sign which can be shown by a filled circle.

• Identify the symbol that correctly describes the relationship.

  Hints

  Take a look at this example:

  $4$ is more than $2$.

  You can write it as $4>2$.

  Or you could write it as $2<4$.

  At least indicates greater than or equal.

  Remember the relation signs:

  • $<$ less than
  • $\le$ less than or equal
  • $>$ greater than
  • $\ge$ greater than or equal

  Solution

  Christopher already knows that his trip takes fewer than 15 days. Fewer than indicates the $<$- or less than-symbol: $<15$. For the representation on a number line you use an empty circle surrounding 15.

  The amount of blood orange he can take with is limited to the top by $1000~ml$, including this amount. This indicates the $\le$- or less than or equal-symbol: $\le 1000$. Here you use a filled circle.

  Christopher knows how much hair gel he needs. So he concludes to pack more than one bottle of hair gel. This indicates the $>$- or greater than-symbol: $>1$. Again you use an empty circle.

  Last but least he packs some capes: at least one for each day and one in spare, 16 in total. At least indicates the $\ge$- or greater than or equal-symbol: $\ge 16$. Here you use a filled circle on a number line.

  But, what's that? Arriving in the blood orange garden Christopher discovers a sign: Sorry! No blood oranges.

• Determine the corresponding mathematical inequality to match the number line.

  Hints

  This number line represents the inequality $x\le 7$.

  • The arrow to the left indicates $<$ or $\le$.
  • The filled circle stands for less than or equal.

  This number line stands for $x>-20$.

  The $\ge$-symbol can be excluded because the empty circle.

  Solution

  First, we consider number lines in general.

  • An arrow to the left indicates a $<$ or $\le$ relation.
  • An arrow to the right indicates a $>$ or $\ge$ relation.

  You can decide if you have to use $<$ or $\le$ respectively $>$ or $\ge$ depending on the circle.

  • An empty circle means $<$ or $>$ depending on the direction of the arrow.
  • A filled circle means $\le$ or $\ge$.

  So we are able to determine the corresponding inequality from left to the right:

  • $x\le 8$
  • $x<8$
  • $x>4$
  • $x\ge 4$

• Examine the inequality according to different word problems.

  Hints

  At least indicates greater than or equal.

  More than indicates the $>$-symbol.

  Distinguish between less than ($<$) and less than or equal ($\le$).

  Solution

  You can memorize different keywords which indicate the regarding inequality-symbol:

  • More than indicates greater than $>$.
  • At least indicates greater than or equal $\ge$.

  Speed limit A speed limit doesn't indicate that you have to drive faster than this limit, so we get the inequality $x\le 45$.

  Birthday party You like to invite less than 10 friends. So we get $x<10$.

  Earphones More than indicates $>$-symbol. This gives us $x>25$.

• Decide which inequality symbol to use.

  Hints

  Pay attention

  • $5<7$ but
  • $-5>-7$

  If you change the sign of the numbers, you also have to change the inequality symbol.

  Take care of the use of $>$ or $\ge$:

  • $7\ge 7$ but $7\not > 7$
  • $7>4$ as well as $7\ge 4$

  Solution

  You can imagine the greater than symbol as a mouth.

  The bigger number eats the larger one.

  • For example $4>2$. You can also use the $\ge$ sign.
  • Similarly $2<4$ as well as $2\le 4$. The order is changed.

  Pay attention to the sign of the numbers:

  • $-2>-4$ as well as $-2\ge -4$.
  • The other way round we can conclude $-4<-2$ or $-4\le -2$.

• Determine the corresponding inequality.

  Hints

  Pay attention to the circle:

  • Empty circles indicate $>$ or $<$.
  • Filled circles indicate $\ge$ or $\le$.

  A circle indicates the number you have to use in the inequality.

  The circle indicates at the one hand the number $65$ and on the other that $65$ belongs to the inequality.

  The arrow to the right indicates $>$ or $\ge$.

  Together we can conclude the following inequality for this number line:

  $x\ge 65$.

  Solution

  You use number lines to represent inequalities.

  First, you draw a circle exactly at the position of the regarding number.

  Depending on the inequality symbol, the circle is filled or empty:

  • Empty: $>$ or $<$
  • Filled: $\ge$ or $\le$

  You can decide $<$ or $\le$ respectively $>$ or $\ge$ depending on the direction of the arrow.

  Here you see four different number lines, top down:

  • $x>-6$
  • $x\le -2$
  • $x<4$
  • $x\ge 2$