Graphing & Writing Inequalities

What is an Inequality? > < > < Inequality – a comparison of two expressions > – greater than < – less than > – greater than or equal to < – less than or equal to

Equation vs. Inequality? Equation – uses an equal sign; typically only one solution Inequality – uses an inequality symbol; infinite amount of solutions

Open Circle vs. Closed Dot Hole (open circles) – are found when there isn’t an “equal” involved in the sign > < Dot (closed circles) – include that number that the dot in on (is an “equal” involved in sign) > <

Graphing Inequalities Inequality Word Sentence Graph x < 3 x is less than 3 0 1 2 3 4 Explanation: 3 is not included but all numbers less than 3 ARE included x < 3 x is less than or equal to 3 0 1 2 3 4 Explanation: 3 IS included and all numbers less than 3 are also included

Graphing Inequalities Inequality Word Sentence Graph x > 3 x is greater than 3 0 1 2 3 4 Explanation: 3 is not included but all numbers greater than 3 ARE included x > 3 x is greater than or equal to 3 0 1 2 3 4 Explanation: 3 IS included and all numbers greater than 3 are also included

Graphing Inequalities Graph each inequality: x > -2 -5 -4 -3 -2 -1 0 1 2 3 4 5 4 > a -5 -4 -3 -2 -1 0 1 2 3 4 5

Graphing Inequalities Graph each inequality: y < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 t < 0 -5 -4 -3 -2 -1 0 1 2 3 4 5

Graphing Inequalities Graph each inequality: m > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 t < 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

Writing Inequalities Word Sentence Inequality w > 6 x > 9 w is more than 6 x is at least 9 x > 9 k is at most -7 k < -7 g is less than 0 g < 0 8 is greater than j 8 > j

Adding & Subtracting Inequalities We add and subtract inequalities just like equations – the inequality is like the equal sign d - 7 < -2 + 7 +7 d < 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

Adding & Subtracting Inequalities We add and subtract inequalities just like equations – the inequality is like the equal sign 9 + a > 11 -9 -9 a > 2 -5 -4 -3 -2 -1 0 1 2 3 4 5

Adding & Subtracting Inequalities We add and subtract inequalities just like equations – the inequality is like the equal sign 1 + s < 5 -1 -1 s < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5

Adding & Subtracting Inequalities We add and subtract inequalities just like equations – the inequality is like the equal sign 2 > n - 1 +1 + 1 3 > n -5 -4 -3 -2 -1 0 1 2 3 4 5

Adding & Subtracting Inequalities We add and subtract inequalities just like equations – the inequality is like the equal sign 3 + b < 1 -3 -3 b < -2 -5 -4 -3 -2 -1 0 1 2 3 4 5

Solving Inequalities Multiplying & Dividing

Multiplying & Dividing Inequalities We multiply and divide inequalities just like equations – the inequality is like the equal sign One MAJOR exception: When multiplying or dividing by a negative number, the inequality flips

Multiplying & Dividing Inequalities Why does the inequality flip? Example: 2 > -3 2 > -3 Multiply each side by a -1 for example x -1 x -1 -2 > 3 -2 is not greater than 3 so we have to flip the sign -2 < 3

Multiplying & Dividing Inequalities Why does the inequality flip? Example: 8 > 6 8 > 6 Divide each side by a -2 for example ÷ -2 ÷ -2 -4 > -3 -4 is not greater than -3 so we have to flip the sign -4 < -3

Multiplying & Dividing Inequalities Examples: m > -5 14 m > -5 14 (14) (14) Multiply each side by 14 m > -70

Multiplying & Dividing Inequalities Examples: 15n > 450 15n > 450 Divide each side by 15 15 n > 30

Multiplying & Dividing Inequalities Examples: -3a < 12 -3a < 12 Divide each side by -3 (notice we are dividing by a “negative” – sign will flip. -3 a > -4

Multiplying & Dividing Inequalities Examples: x < 1 -2 x < 1 -2 (-2) (-2) Multiply each side by -2 – notice we are multiplying by a “negative” so the sign will flip x > -2

Multiplying & Dividing Inequalities Practice: r < 3 -6 r > -18 -15s > 30 s < -2 8h > 32 h > 4 x < 10 2 x < 20

Inequalities Problem Solving Chico is saving for new shoes that cost $87. He already has $9 saved, and he will save the same amount each week. Chico wants to but the shoes in 6 weeks. Use the inequality below to determine the least amount Chico can save each week and still buy the shoes in 6 weeks. 9 + 6x ≥ 87 A.) $9 C.) $15 B.) $13 D.) $16