10.01 Linear Inequalities
State whether the following inequalities are represented using a solid line or a dashed line on a graph:
y - 3 x > 5
x \geq 4
- 2 y + x \leq - 10
y + 3 x > 7
- y - 6 x > 0
y + 3 x \leq - 10
y \geq - 2
3 y + 3 x > 4
Write the inequality that describes the following shaded regions:
Is \left(6, 20\right) a solution of y\geq 4 x + 5?
Is \left(3, 2\right) a solution of 3 x + 2 y \geq 12 ?
Determine which point satisfies each of the following inequalities:
Describe the three steps required to graph the inequality y + 7 x \geq 5.
Consider the inequality y \lt- 3 x - 6.
Solve for the x and y-intercepts of the line y = - 3 x - 6.
Determine which of the following points satisfy the inequality y \lt- 3 x - 6:
Hence sketch the graph of y \lt - 3 x - 6.
Do the points on the line satisfy the inequality?
Consider the inequality y \leq - 2 x + 2.
Solve for the x and y intercepts of the line y = - 2 x + 2.
Determine which of the following points satisfies the inequality y \leq - 2 x + 2:
Hence sketch the graph of y \leq - 2 x + 2.
Do the points on the line satisfy the inequality?
The solutions to the equation y = x - 2 are plotted on the following graph:
Graph the solutions to y \gt x - 2.
Graph the solutions to y \leq x - 2.
Sketch the graphs of the following inequalities:
y \gt 2x - 4
y \leq 3 x - 4
y \lt 3x - 9
y \geq - 2x + 4
y \gt 2x - 4
y \leq x + 5
State the linear inequality that satisfies the following:
A linear inequality has solutions of \left(0, - 5 \right) and \left( - 1 , - 9 \right) on its boundary line and is not satisifed by \left(1, 3\right).
A linear inequality has solutions of \left(0, - 6 \right) and \left( - 6 , - 24 \right) on its boundary line and is satisifed by \left( - 3 , - 10 \right).
Luigi has several bottles of lemonade and Pepsi that he wants to take to a picnic, but he only has enough room in his backpack for 9 bottles.
Let x represent the number of bottles of lemonade that he takes to the picnic.
Let y represent the number of bottles of Pepsi that he takes to the picnic.
Write an inequality to represent the situation.
Graph the inequality on a cartesian plane.
Use the graph to determine whether Luigi can take the following combinations of bottles to the picnic:
1 bottle of lemonade and 11 bottles of Pepsi
3 bottles of lemonade and 11 bottles of Pepsi
3 bottles of lemonade and 4 bottles of Pepsi
5 bottles of lemonade and 3 bottles of Pepsi
Airline passengers are told to be at the gate no less than 15 minutes before a flight's departure. It takes Gwen 30 minutes to get to the airport, and y minutes to go through check in and security, depending on how busy the airport is.
Form an inequality relating x and y, where x represents the number of minutes left until Gwen's flight departs.
Graph the inequality on a cartesian plane.
If the check-in and security process takes 20 minutes, how many minutes before her flight's departure can Gwen leave her house at the latest?
Roxanne has set aside \$14.00 in her shopping budget for fruit this week. Currently, oranges are on sale for \$1.40 each, while apples are on sale for \$1.75 each.
Let x represent the number of oranges that Roxanne buys.
Let y represent the number of apples that Roxanne buys.
Write an inequality to represent the situation.
Graph the inequality on a cartesian plane.
Use your graph to determine whether Roxanne can buy the following combination of fruits:
6 oranges and 7 apples
3 oranges and 4 apples
7 oranges and 4 apples
8 oranges and 1 apples