Solving Algebraically
Solve the inequality: 3x−15>9
x>8
Is x=−4 a solution to the inequality −3x+5≤12
Yes
State the property of inequality applied when solving m+10<15 to get m<5.
The Multiplication Property of Inequality
Solve the inequality: 9m−14<4m+6
m<4
A rental car costs 50 upfront plus 0.15 per mile (m). If your budget for the car is at most $125, write the inequality.
50+0.15m≤125
Solve the inequality: 2(4y+3)≤−10
y≤−2
Write the algebraic inequality for a graph shown with an open circle at w=2.5 and shaded to the right
w>2.5
Identify the solution set type (one solution, no solution, or infinite solutions) resulting from simplifying 2(x+3)>2x−5
Infinite number of solutions
Solve the inequality: −3(2r+1)≥−6r+15
No solution
A teacher wants to spend no more than $300 on supplies. If 15 is reserved for other supplies and books (b) are priced at 18 each, write the inequality and determine the maximum number of books she can buy.
Inequality: 15+18b≤300
Maximum books: 15
Solve the inequality: 5(k−1)+2k≥16
k≥3
Does p=7 satisfy the inequality 10−4p<−18? Justify your answer.
No
Explain the difference between an open circle and a closed circle on a number line when graphing an inequality solution.
A closed circle (≤ or ≥) indicates that the boundary value is included in the solution set, while an open circle (< or >) indicates that the boundary value is not included in the solution set
Solve the inequality: (1/2)h−5≤3h+5
h≥−4
A food bank needs to collect more than 1,000 pounds of food. They currently have 280 pounds. If volunteers collect 60 pounds per week (w), write the inequality and interpret the minimum whole number of weeks required to reach the goal
Inequality: 280+60w>1000.
Minimum of 13 weeks
Solve the inequality: −3(x−5)+2x>4(x+1)−10
x<4.2 (or x<21/5)
If the solution to an inequality is k≥−1.5, what is the nearest whole number that will satisfy this condition?
0
Explain why solving the inequality −0.5y+10<12 requires you to reverse the inequality symbol, citing the specific property used.
Divide by a negative number
Solve the inequality: 7−2(3p+1)≤13−6p.
All Real Numbers (infinite solutions)
A phone plan charges a base fee of 40 plus 0.05 per text message (t). If a user wants their bill to be no greater than $75.50, write the inequality and calculate the maximum number of text messages allowed.
Inequality: 40+0.05t≤75.50
Maximum text messages: 710 texts
Solve the inequality: (2/3)(6x−9)−(1/2)(4x+6)>5
x>7
The final solution to an inequality is x≥4. Describe the algebraic result when verifying a value of x=3 and explain how this justifies the correct graphical boundary.
x=3 can't be right since it is not greater than 4.
Describe two key differences between the solution set representation of a linear equation in one variable (e.g., 3x=12) and a linear inequality in one variable (e.g., 3x>12).
Number of solutions and boundary point inclusion
Solve the inequality: 9(2x−1)−10x≥5(x+2). Express your final boundary point as an improper fraction.
x≥19/3
A lemonade stand spent 30 on supplies. They charge 1.50 per cup (c). A friend sells twice as many cups for 2.00 each. If the combined net profit must be at least , write and solve the inequality for the minimum whole number of cups (c) you must sell.
Inequality: 1.50c+2.00(2c)−30≥100
OR
5.5c−30≥100.
Minimum cups: 24 cups.