Let x be any real number. Write and graph: x is more than 18. Label the endpoint and shade the solution ray.
x > 18; open circle at 18, shade right
“more than” means >; exclude the endpoint.
Day 101Endpoints and Infinite Solutions
Standards 6.EE.B.8
LEARNING TARGET
I can write and graph a constraint using an inequality.
01 · LEARN
Build the idea
Use > or < for a strict bound and ≥ or ≤ when the endpoint is included. Graph a strict bound with an open circle; use a closed circle for an included endpoint. Shade the direction of all solutions.
Let x be any real number. Write and graph: x is at least 10. Give one included value and explain whether the endpoint is included.
x ≥ 10; closed circle at 10, shade right
“at least” means ≥; include the endpoint. Example solution: 11.
02 · PRACTICE
Your turn
Work in order. Start with the foundations, then build toward explanation and application.
Let x be any real number. Write and graph: x is more than 21. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is at least 28. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is at least 16. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is less than 21. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is less than 15. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is at least 20. Label the endpoint and shade the solution ray.
Let x be any real number. Write and graph: x is at most 15. Give one included value and explain whether the endpoint is included.
Let x be any real number. Write and graph: x is at most 18. Give one included value and explain whether the endpoint is included.
A museum group must have more than 10 participants. Let p be the participant count. Write an inequality and graph its real-number solution set. State which solutions make sense as participant counts.
A museum group must have at least 20 participants. Let p be the participant count. Write an inequality and graph its real-number solution set. State which solutions make sense as participant counts.
03 · CHECK
Answers and reasoning
Reveal the teacher answer key
- x > 21; open circle at 21, shade right
“more than” means >; exclude the endpoint.
- x ≥ 28; closed circle at 28, shade right
“at least” means ≥; include the endpoint.
- x ≥ 16; closed circle at 16, shade right
“at least” means ≥; include the endpoint.
- x < 21; open circle at 21, shade left
“less than” means <; exclude the endpoint.
- x < 15; open circle at 15, shade left
“less than” means <; exclude the endpoint.
- x ≥ 20; closed circle at 20, shade right
“at least” means ≥; include the endpoint.
- x ≤ 15; closed circle at 15, shade left
“at most” means ≤; include the endpoint. Example solution: 14.
- x ≤ 18; closed circle at 18, shade left
“at most” means ≤; include the endpoint. Example solution: 17.
- p > 10; open circle at 10, shade right; participant counts are whole numbers 11, 12, …
“more than” means >. The real-number ray has infinitely many solutions, but participant counts must be whole numbers.
- p ≥ 20; closed circle at 20, shade right; participant counts are whole numbers 20, 21, …
“at least” means ≥. The real-number ray has infinitely many solutions, but participant counts must be whole numbers.