1.4 Write Equations And Inequalities Algebra I 1.4 Write Equations And Inequalities

VOCAB Equation – a mathematical sentence formed by placing the symbol = between two expressions Inequality – a mathematical sentence formed by placing one of the symbols <, >, ≤, or ≥ between two expressions

VOCAB Open Sentence – an equation or inequality that contains an algebraic expression Solution to an Equation –a number that makes the sentence true Solution to an Inequality – a number or set of numbers that makes the sentence true

1.4 Write Equations and Inequalities Symbol Meaning Associated Words = is equal to The same as < is less than Fewer than > is greater than More than ≤ is less than or equal to At most; no more than ≥ is greater than or equal to At least; no less than

1.4 Write Equations and Inequalities The BIG Difference Equations: Have ONLY ONE Solution! Inequalities: Have MANY Solutions!!!!!

EXAMPLE 1 Write equations and inequalities Verbal Sentence Equation or Inequality a. The difference of twice a number k and 8 is 12. 2k – 8 = 12 b. The product of 6 and a number n is at least 24. 6n ≥ 24

c. A number y is no less than 5 and no more than 13. GUIDED PRACTICE for Example 1 c. A number y is no less than 5 and no more than 13. 5 ≤ y ≤ 13 d. Write an equation or an inequality: The quotient of a number p and 12 is at least 30. ANSWER P 12 > – 30

Check possible solutions EXAMPLE 2 Check possible solutions Check whether 3 is a solution of the equation or inequality. Equation/Inequality Substitute Conclusion a. 8 – 2x = 2 8 – 2(3) 2 ? = 2 = 2 3 is a solution.  b. 4x – 5 = 6 4(3) – 5 6 ? = 7 = 6 3 is not a solution. X c. 2z + 5 > 12 2(3) + 5 12 > ? 11 > 12 3 is not a solution. X d. 5 + 3n ≤ 20 5 + 3(3) 20 ≤ ? 14 ≤ 20 3 is a solution. 

GUIDED PRACTICE for Example 2 Check to see whether or not 5 is a solution of the equation or inequality. Equation/Inequality Substitute Conclusion a. 9 – x = 4 9 – 5 4 ? = 4 = 4 5 is a solution.  b. b + 5 < 15 5 + 5 15 < ? 10<15 5 is a solution.  c. 2n + 3 21 > – 2(5) + 3 21 > – ? 13 21 5 is NOT a solution. > – X

EXAMPLE 3 Use mental math to solve an equation Equation Think Solution Check a. x + 4 = 10 What number plus 4 equals 10? 6 6 + 4 = 10  b. 20 – y = 8 20 minus what number equals 8? 12 20 –12 = 8  c. 6n = 42 6 times what number equals 42? 7 6(7) = 42  a 5 = 9 d. 45 5 = 9  What number divided by 5 equals 9? 45

GUIDED PRACTICE for Example 3 Solve the equation using mental math. Equation Think Solution Check 5. m + 6= 11 What number plus 6 equals 11? 5 5 + 6 = 11  6. 5x = 40 5 times what number equals 40? 8 5(8) = 40  7. r = 10 4 What number divided by 4 equals 10 40 40 = 10  4

Is 2 a solution to 4z – 5 < 3? A solution NOT a solution Answer Now

Solve for f: 12 24 4 Answer Now

EXAMPLE 4 Solve a multi-step problem Mountain Biking The last time you and 3 friends went to a mountain bike park, you had a coupon for $10 off and paid $17 for 4 tickets. What is the regular price of 4 tickets? If you pay the regular price this time and share it equally, how much does each person pay?

Step 1: Write a verbal model. EXAMPLE 4 Solve a multi-step problem Step 1: Write a verbal model. Let p be the regular price of 4 tickets. Write an equation. Regular Price – Coupon = Amount Paid P – 10 = 17

Step 2: Use mental math to solve the equation p – 10 = 17. EXAMPLE 4 Solve a multi-step problem Step 2: Use mental math to solve the equation p – 10 = 17. Think: 10 less than what number is 17? Because 27 – 10 = 17, the solution is 27. Answer: The regular price for 4 tickets is $27. Step 3: Find Cost Per Person $27 / 4 people = 6.75 Answer: $6.75 per person.

GUIDED PRACTICE for Examples 4 and 5 WHAT IF? Suppose that the price of 4 tickets with a half-off coupon is $15. What is each person’s share if you pay full price?

STEP 1: Write a verbal model. GUIDED PRACTICE for Examples 4 and 5 STEP 1: Write a verbal model. Let p be the regular price of 4 tickets. Write an equation. Regular Price – Coupon = Amount Paid r – 15 = 15

STEP 2: Use mental math to solve the equation p – 15=15. GUIDED PRACTICE for Examples 4 and 5 STEP 2: Use mental math to solve the equation p – 15=15. Think: 15 less than what number is 15? Because 30 – 15 = 15, the solution is 30. So the full price is $30. STEP 3: Find the Cost Per Person $30/4 = 7.5 Answer: $7.50 per person

Write and check a solution of an inequality EXAMPLE 5 Write and check a solution of an inequality You play 18 games of basketball. Last year you scored over 351. What was the average number of points you earned for each game. STEP 1: Write a verbal model. Let p be the average number of points per game. Write an inequality. Number of Games • Number of Points Per Game > Total Points Last Year 18 • p > 351 STEP 2: Check that 20 is a solution of the in equality18p > 351. 18(20) = 360 360 > 351 Answer: An average of 20 points per game will be enough.

GUIDED PRACTICE for Examples 4 and 5 WHAT IF Suppose that the player plays 16 games. Would an average of 22 points per game be enough to beat last year’s total? STEP 1: Write a verbal model. Let p be the average number of points per game. Write an inequality. Number of Games • Number of Points Per Game = Total Points Last Year STEP 2: Check that 22 is a solution of the in equality16p > 351. Because 16(22) = 352 352 > 351 So, 22 is a solution.