Solving Inequalities



Algebra 1

7.1 Solving 1 and 2 step inequalities.

NOTES

Recall:

< is read as “Less than” or “fewer than”

> is read as “Greater than” or “more than”

[pic] is read as “Less than or equal to”; [pic] is read as “Greater than or equal to”

Solving an inequality is similar to solving an equation.

Just like when we solve equations, our goal is to isolate the variable on one side of the inequality.

Solving and graphing one-step inequalities:

• Use the additive inverse to get the ‘x’ alone.

• Bring down the inequality symbol

• Graph the solution on a number line. Be sure to check if the endpoint of the graph should be an open circle or a dot. (closed circle)

• If inequality is ≤ or ≥, the endpoint should be included and a dot (closed circle) should be used.

• If inequality is < or >, the endpoint should not be included and an open circle should be used.

• Shade the side of the number line that the solutions belong to.

HINT: If the variable is in front of the inequality symbol, the symbol points the direction you should shade.

A. B.

x - 12 > 8 x + 13 < 19

+ 12 +12 - 13 -13

x > 20 x < 6

A. The solution is “x is greater than 20” which means that any number greater than 20 is a solution. In an equation there is exactly 1 answer. In an inequality there are infinite solutions.

B. The solution is “x ________________ than 6”

What is one number that could be a solution to this inequality? __________

To solve a one step inequality with multiplication or division:

• Use the multiplicative inverse to get the ‘x’ alone.

• When multiplying or dividing by a positive number, just bring down the inequality symbol.

• When multiplying or dividing by a negative number, FLIP the inequality symbol.

• Graph the solution on a number line.

C. 5x ≥ 15 D. -2x ≥ 14 E. [pic]

5x ≥ 15 -2x ≤ 14 [pic] (flip)

5 5 -2 -2 x ≥ -6

x ≥ 3 x ≤ - 7

Two step inequalities are solved exactly the same way, they just have one additional step!

• Use the Additive inverse to remove the term that is adding or subtracting the x term.

• Use the Multiplicative inverse to isolate the variable. Remember: If you multiply or divide by a negative number, FLIP the inequality symbol!

• Graph the solution on a number line.

PRACTICE PROBLEMS

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

ALGEBRA 1 – ASSIGNMENT Name: _______________________

7.1 Solving and Graphing Inequalities

Solve and graph each inequality. C-level

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

Write an inequality describing each graph below.

7. 8.

9. 10.

Solve and graph each inequality. B-level

11. [pic] 12. [pic]

13. 2[pic] 14. [pic]

15. [pic] 16. [pic]

Solve and graph each inequality. A-level

17. [pic] 18. [pic]

19. [pic] 20. [pic]

21. In Buffalo, NY, winter temperatures often do not exceed [pic] Celsius. Write an inequality that describes the January temperatures, T, in Buffalo.

22. Jerry has only $27 to spend on school supplies. He spends $18 on a backpack. Write an inequality that models how much more Jerry can spend on school supplies and stay within his limit.

23. A school auditorium can seat 450 people for graduation. The graduates will use 74 seats. Write and solve an inequality to describe the number of additional people who can be seated in the auditorium.

PRACTICE TEST

|Level C |Level B |Level A |

|1C. Solve and graph the inequality. |1B. Solve the inequality and graph. |1A. Use the inequality below to answer the |

| | |questions. |

|8x < 24 |        [pic] | |

| | |        [pic] |

| | |     [pic]       |

| | |a) Solve the inequality. |

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| | |b) Graph the solution. |

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| | |c) Write 3 solutions for the inequality. |

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ANSWER KEY

|1. [pic]x > 2 |2. n < 4 |3. y < -3 |

| | | |

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|4. [pic] |5. [pic] |6. [pic] |

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|7. [pic] |8. x < -3 |9. x > -2 |

|10. [pic] |11. n < -2 |12. n > 5 |

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|13. [pic] |14. y < 2 |15. [pic] |

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|16. n > 6 |17. x > -2 |18. x < -5 |

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|19. [pic] |20. [pic] |21. [pic] |

| | |22. [pic] |

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| | |23. [pic] |

|Practice Test | |1A |

|1C x < 3 |1B [pic] |a) x > 3 |

| | |b) |

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| | | |

| | |c) Answers vary |

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