1.5 – Solving Inequalities

1.5 – Solving Inequalities

*Just like solving equations*

1.5 – Solving Inequalities

*Just like solving equations*

Example 1

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x >

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

1.5 – Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

1.5– Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

0

1.5– Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

0 1

1.5– Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

0 1

1.5– Solving Inequalities

*Just like solving equations*

Example 1 - Solve 7x – 5 > 6x – 4 and graph on a number line.

7x – 5 > 6x – 4

-6x -6x

x – 5 > -4

+5 +5

x > 1

0 1

Example 2

Example 2 – Solve –m < -m + 4 and graph. 9

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m <

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m <

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m < 4

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m < 4

-8 -8

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m < 4

-8 -8 m

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m < 4

-8 -8 m >

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 9

9·–m < -m + 4 ·9 9

–9m < -m + 4 +m +m –8m < 4

-8 -8 m >

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 99·–m < -m + 4 ·9 9–9m < -m + 4 +m +m –8m < 4

-8 -8 m > -½

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 99·–m < -m + 4 ·9 9–9m < -m + 4 +m +m –8m < 4

-8 -8 m > -½

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 99·–m < -m + 4 ·9 9–9m < -m + 4 +m +m –8m < 4

-8 -8 m > -½

0

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 99·–m < -m + 4 ·9 9–9m < -m + 4 +m +m –8m < 4

-8 -8 m > -½

0-1½ -1 -½

Example 2 – Solve –m < -m + 4 and graph. 9

–m < -m + 4 99·–m < -m + 4 ·9 9–9m < -m + 4 +m +m –8m < 4

-8 -8 m > -½

0-1½ -1 -½

Example 3

Example 3 – Define a variable and write an inequality for each problem. Then solve.

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x >

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x >

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x >

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x > 35

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x > 35

-4 -4

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x > 35

-4 -4

x <

Example 3 – Define a variable and write an inequality for each problem. Then solve.

The product of –4 and a number is at least 35.

-4x > 35

-4 -4

x < -8.75

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) +

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) +

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) + 25%( )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) + 25%( )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) + 25%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%( ? ) + 25%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%( ? )

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x)

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x)

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76)

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) +

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x)

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x) ≥ 80

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x) ≥ 80

57

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x) ≥ 80

57 + 0.25x

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 80 57 + 0.25x ≥ 80

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 80 57 + 0.25x ≥ 80

-57 -57

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x) ≥ 80

57 + 0.25x ≥ 80

-57 -57

0.25x

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 80

0.75(76) + 0.25(x) ≥ 80

57 + 0.25x ≥ 80

-57 -57

0.25x ≥ 23

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 8057 + 0.25x ≥ 80

-57 -57 0.25x ≥ 23 0.25 0.25

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 8057 + 0.25x ≥ 80

-57 -57 0.25x ≥ 23 0.25 0.25

x

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 8057 + 0.25x ≥ 80

-57 -57 0.25x ≥ 23 0.25 0.25

x ≥

Ex. 4 The final grade for a class is calculated by taking 75% of the average test score and adding 25% of the score on the final exam. If all scores are out of 100 and a student has a 76 test average, what score does the student need to make on the final exam to have a final grade of at least 80?

75%(76) + 25%(x) ≥ 800.75(76) + 0.25(x) ≥ 8057 + 0.25x ≥ 80

-57 -57 0.25x ≥ 23 0.25 0.25

x ≥ 92