Inequalities and Their Graphs

Inequalities and Their Graphs exercise

• Describe the process of graphing inequalities.

  Hints

  Any greater than or less than inequality will be graphed with an open circle. This indicates the endpoint value is not included in the solution set.

  Any greater than or equal to or less than or equal inequality will be graphed with a closed circle to include the end point value in the solution set.

  Solution

  Remember, we graph less than inequalities with an open circle and an arrow pointing to the left. Since Timmy must obey the $55$ mph speed limit, he may drive at any speed less than $55$ mph. Our inequality is: $x < 55$.

  We graph less than or equal to inequalities with a closed circle and an arrow pointing to the left. Because Timmy can drive exactly $55$ mph, we use a closed circle to include $55$ in our solution. Timmy can also drive slower than $55$ mph, so any value to left of $55$ on the number line will also be included as a part of our solution set. We write this inequality as: $x \leq 55$.

  To graph this greater than inequality, we again use an open circle on $55$. This time, however, our arrow will point to the right to include any speed above $55$. The inequality to express this is: $x > 55$.

• Explain the meaning of the graphs.

  Hints

  Remember, when graphing less than or greater than inequalities, use an open circle on the number line to show that the value is not a solution.

  When graphing less than or equal to or greater than or equal to inequalities, use a closed circle on the number line to include the value in the solution set.

  Solution

  The graph of $x < 55$ will have an open circle at $55$ and an arrow pointing to the left to include all numbers less than $55$.

  The graph of $x > -20$ will have an open circle at $-20$ and arrow pointing to the right to include all numbers greater than $-20$.

  The graph of $x \leq -20$ will have a closed circle at $-20$ and an arrow pointing to the left to show that $-20$ as well as all numbers less than $-20$ are in the solution set.

  The graph of $x \leq 55$ will have a closed circle at $55$ and an arrow pointing to the left to show that $55$ as well as all numbers less than $55$ are in the solution set.

• Determine all speeds that will not result in a speeding ticket.

  Hints

  An open circle on the number line excludes a value from the possible solution set. A closed circle includes the value.

  Consider the direction of the arrows. Normally, road signs showing speed limits post a maximum but not a minimum speed.

  Solution

  Of course Timmy wants to be on time, but he doesn't want to receive a speeding ticket. So the question is, how fast can he drive? The road sign shows $65$ mph, so all speeds less than or equal to $65$ are allowed. The graph that represents this situation is on the right. There are also two other graphs that show speed less than $65$.

  HURRY, TIMMY!!!

• Match the correct inequality with the graph.

  Hints

  Remember, when graphing less than or greater than inequalities, use an open circle to show that the solution set does not include the value.

  When graphing less than or equal to or greater than or equal to inequalities, use a closed circle to show that the solution set does include the value.

  Solution

  To graph $x \leq 5$ on the number line, use a closed circle to include $5$ in the solution set and an arrow pointing to the left to include all numbers less than $5$.

  The inequality $x < -15$ is graphed with an open circle at $-15$ and an arrow pointing to the left to include all numbers less than $-15$, but not $-15$.

  We graph $x \geq -10$ with a closed circle at -10 and an arrow to the right.

  Lastly, we graph the inequality $x> 18$ with an open circle at 18 and an arrow to the right.

• Translate the words into a mathematical sentence using inequality signs.

  Hints

  Remember: the open side of the inequality sign points towards the larger quantity. The closed side points towards the smaller quantity.

  $3>2$ is a true statement as well as $3 \ge 2$.

  Solution

  The greater than and less than symbols help us to determine relative size between different quantities.

  • When $x$ is greater (larger) than $y$, we use the $x>y$ sign.
  • When $x$ is greater than or equal to $y$, we use the $x \ge y$ sign.
  • When $x$ is less (smaller) than $y$, we use the $x <y$ sign.
  • When $x$ is less than or equal to $y$, we use the $x \le y$ sign.

• Describe each situation using inequalities.

  Hints

  Keep in mind that a greater than or less than inequality does not include the value in the solution set. If a ride requires that riders are taller than a particular height, anyone of that exact height would be considered too short to ride.

  Different rides require that riders are a certain height or taller or are a certain height or shorter; riders who are the exact height would be able to go on these rides. These are examples of a greater than or equal to or a less than or equal to inequality, so the stated height values would be included in the solution sets.

  Solution

  For the Crazy Eight Roller Coaster, the inequality is $x > 4.5$. The Crazy Eight requires that riders are taller than $4.5$ feet, but not $4.5$ feet exactly. Because Kayla is exactly $4.5$ feet tall, she cannot ride the Crazy Eight because the inequality doesn't include $4.5$ in the solution set. Kayla's height is therefore less than the height required for the Crazy Eight.

  The inequality for the Endless Loopster can be written as the inequality: $x \geq 4$. We know that Kalya is $4.5$ feet tall and that Tanisha is $4.75$ feet tall. Both girls can ride the Endless Loopster because both girls are taller than $4$ feet.

  To ride the Sky High Ferris Wheel, riders have to be shorter than $6.7$ feet tall. We can write this inequality as: $x < 6.7$. Kayla and Tanisha are both shorter than $6.7$ feet tall , so they can ride the Sky High Ferris Wheel.