Algebra I 1.4 Write Equations And Inequalities. VOCAB Equation – a mathematical sentence formed by placing the symbol = between two expressions Inequality.

Algebra IAlgebra I

1.4 Write Equations 1.4 Write Equations And InequalitiesAnd Inequalities

VOCABEquation – 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

VOCABOpen Sentence – an equation or inequality that contains an algebraic expressionSolution to an Equation –a number that makes the sentence trueSolution to an Inequality – a number or set of numbers that makes the sentence true

1.4 Write Equations and 1.4 Write Equations and InequalitiesInequalities

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 1.4 Write Equations and InequalitiesInequalities

The The BIGBIG Difference Difference

Equations:Equations:Have ONLY ONE Solution!Have ONLY ONE Solution!

Inequalities:Inequalities:Have MANY Solutions!!!!!Have MANY Solutions!!!!!

Write equations and inequalities

EXAMPLE 1

a. The difference of twice a number k and 8 is 12.

b. The product of 6 and a number n is at least 24.

6n ≥ 24

Verbal Sentence Equation or Inequality

2k – 8 = 12

GUIDED PRACTICE for Example 1

d. Write an equation or an inequality: The quotient of a number p and 12 is at least 30.

ANSWER

P12

>– 30

c. A number y is no less than 5 and no more than 13.

5 ≤ y ≤ 13

Check possible solutions

EXAMPLE 2

Check whether 3 is a solution of the equation or inequality.

a. 8 – 2x = 2

c. 2z + 5 > 12

b. 4x – 5 = 6

d. 5 + 3n ≤ 20

4(3) – 5 6?=

8 – 2(3) 2?=

2(3) + 5 12>?

5 + 3(3) 20

≤?

Equation/Inequality Substitute Conclusion

2 = 2 3 is a solution.

14 ≤ 20 3 is a solution.

7 = 6 3 is not a solution.

X

11 > 12 3 is not a solution.

X

GUIDED PRACTICE for Example 2

Check to see whether or not 5 is a solution of the equation or inequality.

a. 9 – x = 4

b. b + 5 < 15

9 – 5 4?=

Equation/Inequality Substitute Conclusion

4 = 4 5 is a solution.

5 + 5 15<?

10<15 5 is a solution.

c. 2n + 3 21>– 13 215 is NOT a solution.

>–2(5) + 3 21>–?

X

EXAMPLE 3Use mental math to solve an equation

Equation

a. x + 4 = 10

b. 20 – y = 8

c. 6n = 42

a5

=9d.

Think

What number plus 4 equals 10?

20 minus whatnumber equals 8?

6 times what numberequals 42?What number divided by 5 equals 9?

Solution

6

12

7

Check

20 –12 = 8

6 + 4 = 10

6(7) = 42

455 =9 45

GUIDED PRACTICE for Example 3

Solve the equation using mental math.

Equation

5. m + 6= 11

6. 5x = 40

Think

What number plus 6 equals 11?

5 times whatnumber equals 40?What number dividedby 4 equals 10

Solution

5

8

40

Check

5(8) = 40

5 + 6 = 11

40 = 10 4

7. r = 104

Is 2 a solution to 4z – 5 < 3?Is 2 a solution to 4z – 5 < 3?

Answer NowAnswer Now

1.1. A solutionA solution

2.2. NOT a solutionNOT a solution

Solve for f:Solve for f:

Answer NowAnswer Now

1.1. 00

2.2. 1212

3.3. 2424

4.4. 44

46

f

Solve a multi-step problemEXAMPLE 4

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?

Mountain Biking

Solve a multi-step problem

EXAMPLE 4

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

Solve a multi-step problemEXAMPLE 4

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?

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

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

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 > 351So, 22 is a solution.