Inequalities (Describing, Compound, Area Inequalities, etc.)
−6 < −2x+4 ≤ 8
Subtract 4 from all parts.
−10 < −2x ≤ 4
Divide all parts by -2 (flip signs!)
5 > x ≥ −2
A = lw (Area of a rectangle)
✅ Solve for w.
Divide both sides by l:
w = A / l
Is the number √16 rational or irrational?
√16 = 4 → Rational
Perfect square example
3x + 5 = 20
x = 5
Problem: The sum of three consecutive integers is 48. What are the integers?
x+(x+1)+(x+2)=48
3x+3=48
3x=45
x=15
Integers: 15, 16, 17
A store sells notebooks for $3 each. You have at most $24.
Write and solve an inequality to find how many notebooks (n) you can buy.
3n ≤ 24
n ≤ 8 "n is all real numbers less than or equal to 8"
✅ You can buy up to 8 notebooks.
C = 2πr (Circumference of a circle)
✅ Solve for r.
Divide both sides by 2π.
r = C / (2π)
Simplify √49 and state if it’s rational or irrational.
√49 = 7 → Rational
(m/3) + 5 = 2
m = -9
Problem: The sum of five consecutive odd integers is 95. What are the integers?
x+(x+2)+(x+4)+(x+6)+(x+8)=95
5x+20=95
5x=75
x=15
Integers: 15, 17, 19, 21, 23
Solve: 2x + 5 > 3x - 4
Subtract 2x.
5 > x - 4
Add 4.
9 > x Variable needs to be on the left!
x < 9 "X is all real numbers less than 9"
A = ½bh — Solve for h.
Multiply both sides by 2.
2A = bh
Divide by b.
h = 2A / b
Which of the following is irrational?
A) 0.25
B) 3/7
C) √2
D) -5
C) √2
5 (z -2 ) = 3z + 6
z = 8
Problem: The sum of four consecutive even integers is 80. Find the integers.
x+(x+2)+(x+4)+(x+6)=80
4x+12=80
4x=68
x=17
A rectangular garden must have an area less than 50 square meters.
The length is 5 meters, and the width is w meters.
Write and solve an inequality to find the possible widths.
A=length × width
5w < 50
w<10
4x - 5 = 4x - 5
Subtract 4x: -5 = -5
♾️ Infinite solutions
Is the number 0.101001000100001... rational or irrational?
❌ Irrational → It does not terminate or repeat.
3(5+pink)+ 4(pink) = 64 -> 15 + 3p + 4p = 64
15 + 7p = 64 -> 7p = 49
p = $7 (pink)
purple = 7 + 5 = $12
12 + 7 = $19
The sum of three consecutive even integers is 54. What are the integers?
x+(x+2)+(x+4)=54
3x+6=54
3x=48
x=16
Integers: 16, 18, 20
x−7 ≤ −2 or 4x > 20
x - 7 ≤ -2
x ≤ 5
4x > 20
x > 5
x ≤ 5 OR x > 5
Solve the inequality:
5x + 3 < 5x - 2
Subtract 5x from both sides: 3 < -2
"3 is less than -2" (false)
❌ No solution
Simplify √81 + √4. Is the result rational or irrational?
√81 = 9, √4 = 2 → 9 + 2 = 11 → ✅ Rational
Logan combines y pounds of brown sugar at $8 per pound with 3 pounds of white sugar at $5 per pound. She makes a mixture that averages $6.25 per pound. Write an equation to model the situation. (Draw a table; use Guided Notes 1-3 )
8y + 15 = 6.25 (y + 3)
The sum of five consecutive integers is 95. What are the integers?
x+(x+1)+(x+2)+(x+3)+(x+4)=95
5x+10=95
5x=85
x=17
17, 18, 19, 20, 21