An inequality is a mathematical sentence that contains inequality symbols such as
>, <, ≥, ≤
between numerical or variable expressions. Four types of simple inequalities and their graphs are shown below.
Verbal Expression :
All real numbers less than 4.
Inequality :
x < 4
Graph :
Verbal Expression :
All real numbers greater than -3.
Inequality :
x > -3
Graph :
Verbal Expressions :
All real numbers less than or equal to 2.
All real numbers at most 2.
All real numbers no greater than 2.
Inequality :
x ≤ 2
Graph :
Verbal Expressions :
All real numbers greater than or equal to -1.
All real numbers at least -1.
All real numbers no less than -1.
Inequality :
x ≥ -1
Graph :
Notice on the graphs that we use an open dot for > or < and a solid dot for ≥ or ≤.
For all real numbers a, b, and c, the following are true.
Transitive Property :
If a < b and b < c, then a < c.
Addition Property :
If a < b, then a + c < b + c.
Subtraction Property :
If a < b, then a - c < b - c.
Multiplication Property :
If a < b and c is positive, then a ⋅ c < b ⋅ c.
Division Property :
If a < b and c is positive, then ᵃ⁄c < ^{b}⁄c.
Note :
Whne you multiply or divide both sides of an inequality by the same negaive number, you have to flip the inequality sign.
Examples :
2 < 3 ----> 2 ⋅ (-1) > 3 ⋅ (-1) ----> -2 > -3
2 < 3 ----> ²⁄₍₋₁₎ > ³⁄₍₋₁₎ ----> -2 > -3
Problem 1 :
(a) The number x is no greater than -2.
(b) The amount of calories n meet or exceed 1,200.
Solution :
Part (a) :
x ≤ -2
Part (a) :
n ≥ 1200
Problem 2 :
Solve for n.
4n - 9 ≥ 12 - 3n
Solution :
4n - 9 ≥ 12 - 3n
Add 3n to both sides.
7n - 9 ≥ 12
Add 9 to both sides.
7n ≥ 21
Divide both sides by 7.
n ≥ 3
Problem 3 :
If -3 + n ≤ 25, which inequality represents the possible range of values for 4n - 12?
(A) 4n - 12 ≤ -100
(B) 4n - 12 ≤ 100
(C) 4n - 12 ≥ -100
(D) 4n - 12 ≥ -100
Solution :
-3 + n ≤ 25
Add 3 to both sides.
n ≤ 28
Multiply both sides by 4.
4n ≤ 112
Subtract 12 from both sides.
4n - 12 ≤ 100
Therefore, the correct answer is option (B).
Problem 4 :
Which of the following numbers is NOT a solution to the inequality (½)x - ⅓ > ⁷⁄₉ + (⁵⁄₂)x?
(A) -⁷⁄₂
(B) -⁵⁄₂
(C) -³⁄₂
(D) -½
Solution :
(½)x - ⅓ > ⁷⁄₉ + (⁵⁄₂)x
Multiply both sides by 2.
2[(½)x - ⅓] > 2[⁷⁄₉ + (⁵⁄₂)x]
2[(½)x] - 2[⅓] > 2[⁷⁄₉] + 2[(⁵⁄₂)x]
x - ⅔ > ¹⁴⁄₉ + 5x
Multiply both sides by 9.
9[x - ⅔] > 9[¹⁴⁄₉ + 5x]
9x - 9(⅔) > 9(¹⁴⁄₉) + 9(5x)
9x - 6 > 14 + 45x
Subtract 9x from both sides.
-6 > 14 + 36x
Subtract 14 from both sides.
-20 > 36x
Divide both sides by 36.
-⁵⁄₉ > x
x < -⁵⁄₉
x < -0.56
In option (D), -½ = -0.5 which is greater than -0.56.
-½ is not a solution to the inequality.
Therefore, the correct answer is option (D).
Problem 5 :
-3a + 7 ≥ 5a - 17
In the inequality shown above, find the greatest possible value of 3a + 7.
(A) 16
(B) 14
(C) 12
(D) 10
Solution :
-3a + 7 ≥ 5a - 17
Add 3a to both sides.
7 ≥ 8a - 17
Add 17 to both sides.
24 ≥ 8a
Divide both sides by 8.
3 ≥ a
a ≤ 3
Multiply both sides by 3.
3a ≤ 9
Add 7 to both sides.
3a + 7 ≤ 16
3a + 7 can be equal to 16 or less than 16.
The greatest possible value of 3a + 7 is 16.
Therefore, the correct answer is option (A).
Problem 6 :
Nine is not more than the sum of a number and 17.
Which of the following inequalities represents the statement above?
(A) 9 ≥ 17x
(B) 9 ≥ x + 17
(C) 9 ≤ 17x
(D) 9 ≤ x + 17
Solution :
Let x be the number.
nine
9
is not more tha n
≤
the sum of x and 17
x + 17
The inequality represents the given statement :
9 ≤ x + 17
Therefore, the correct answer is option (D).
Problem 7 :
The product of 7 and number y is no less than 91.
Which of the following inequalities represents the statement above?
(A) 7y ≤ 91
(B) 7y < 91
(C) 7y ≥ 91
(D) 7y > 91
Solution :
Let x be the number.
the product of 7 and y
7y
is no less than
≥
91
91
The inequality represents the given statement :
7y ≥ 91
Therefore, the correct answer is option (C).
Problem 8 :
Which of the following inequalities represents the graph above?
(A) m ≤ -5
(B) m < -5
(C) m ≥ -5
(D) m > -5
Solution :
In the above graph, it is clear that the value of the variable m is equal to -5 or greater than -5.
m ≥ -5
Therefore, the correct answer is option (C).
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