Solving Inequalities & Problem Solving with Inequalities

Review Inequalities: Less than < x<5 Greater than > x>-2 Less than or equal to ≤ x ≤3 Greater than or equal to ≥ x ≥0

To graph Inequalities: Use an open circle for < or > then draw an arrow and shade the line in for which direction it will go. Use a closed circle for ≤ or ≥ then draw an arrow and shade shade the line in for which direction it will go. Example: x< 4 1 2 3 4 5 6

write and graph each of the following inequalities. The summer temperature T, in Phoenix is at least 80 degrees. The average snow fall is less than one inch. (*use less than but not or equal to) The class average is 85% and higher (*use greater than or equal to) (*use greater than or equal to) T ≥ 80 degrees 77 78 79 80 81 82 S < 1 -2 -1 0 1 2 3 4 A ≥ 85% 82 83 84 85 86 87

Tips to Solving Linear Inequalities: Solve just like an equation Get the variable alone on one side Remember if you multiply or divide by a negative number to get the variable alone, reverse (change) the inequality symbol! -2x < 8 x > -4

Example: 2x + 3 > 4 As an equation 2x + 3 = 4 -3 -3 2x = 1 2 2 x = -3 -3 2x = 1 2 2 x = As an inequality 2x + 3 > 4 - 3 -3 2x > 1 2 2 x > Notice the x value is the same. Now graph your answer. 1 2 3

x – 8 < -10 -3x + 4 > 19 -3x > 15 x < -2 x < -5 -5 -4 -3 -2 -1 0 1 2 3 -3x + 4 > 19 -3x > 15 *flip the > to a < because divided by -3 x < -5 -5 -4 -3 -2 -1 0 1 2 3

Practice: solve and graph x + 3 ≤ 2(x-4) x + 3 ≤ 2x – 8 -x -x 3 ≤ x – 8 +8 +8 11 ≤ x Tip: Same meaning as x > 11 and usually easier to decide which way the arrow points when you write the variable on the left 8 9 10 11 12 13 14 15

In Owego, NY, the temperature in January may not exceed 0 degrees C In Owego, NY, the temperature in January may not exceed 0 degrees C. Write an inequality that describes temperature T for the month and graph it. T ≤ 0 -2 -1 0 1 2 3 4

Problem Solving With Inequalities You have $32.14 in your pocket that you want to spend on a new pair of shoes. If sales tax is 6%, what’s the most your shoes can cost before tax? Label what you know: You have = $32.14 Sales tax = .06 “x” cost of shoes Write an inequality. x + x(.06) ≤ 32.14 Solve 1.06x ≤ 32.14 x ≤ 30.32 Answer the question: The shoes can cost up to $30.32 and you would have enough money to pay for them.

The last problem to solve together: You need some new CD’s to copy your photos on to send to your grandma in St. Louis. You found a great deal where you can get CD’s for $3.00 each plus a flat shipping fee of $2.00. If you have $25.00 to spend, how many CD’s can you order? Label what you know: CDs= $3 Shipping = $2 You have = $25 “x” number of CDs Write an inequality. 3x + 2 ≤ 25 Solve 3x ≤ 23 x ≤ 7 2/3 Answer the question: You can order up to 7 CDs and have enough money to pay for them.