Are the two inequalities equivalent? - 7$x$ < 35 and - $x$ < 5
a.
Yes
b.
No
Solution:
- 7x < 35 [First inequality.]
- x < 5 [Divide throughout by 7.]
- x < 5 [Second inequality.]
If two inequalities have the same solution set, then they are called equivalent inequalities.
So, the two inequalities are equal.
Correct answer : (1)
2.
Are the two inequalities equivalent? 3$x$ - 5 ≤ 13 and 3$x$ ≥ 18
a.
Yes
b.
No
Solution:
3x - 5 ≤ 13 [First inequality.]
3x ≤ 18 [Add 5 to both sides of the equation.]
3x ≥ 18 [Second inequality.]
If the two inequalities have the same solution set, then they are called equivalent inequalities.
So, the given inequalities are not equal.
Correct answer : (2)
3.
Solution for the inequality is given in set-builder notation. Solve and choose the correct one for the inequality given. 7 [2$x$ - ($x$ + 7)] < 3 (2$x$ - 3)
x - 49 < - 9 [Subtracting 6x from the two sides of the equation.]
x < 40 [Add 49 to both sides of the equation.]
The solution set is {x: x < 40}
Correct answer : (3)
4.
Solution for the inequality is given in set-builder notation. Solve and choose the correct one for the inequality given. 3[2$y$ + (3$y$ - 1)] ≥ 5(2$y$ + 1)
5y - 3 ≥ 5 [Subtracting 10y from the two sides of the equation.]
5y ≥ 8 [Add 3 to both sides of the equation.]
y ≥ 8 / 5 [Divide throughout by 5 .]
y ≥ 1.6
The solution set is {y: y ≥ 1.6}
Correct answer : (3)
5.
The difference of the measures of any two sides of a triangle is always less than the measure of the third side. In a triangle ABC, BC = 6 and AC = 4 + AB. Then measure of AB will be:
a.
Less than 1
b.
Greater than 1
c.
Equal to 1
Solution:
The difference of the measures of any two sides of a triangle is always less than the measure of the third side.
So, BC - AC < AB
6 - (4 + AB) < AB [Substitute the values.]
2 - AB < AB
2 < 2AB [Add AB to both sides of the equation.]
1 < AB [Divide throughout by 2.]
So, AB > 1
Correct answer : (2)
6.
Katie can spend at most $788.10 on a television, which includes 6.5% sales tax. Find the maximum price of the television which she can buy.
a.
$788.10
b.
$106.50
c.
$704
d.
$740
Solution:
Let x be the maximum price of the television.
So, sales tax = 6.5x100 [Sales Tax = 6.5%.]
She can afford a maximum of $788.10.
x + 6.5x100 ≤ 788.10 [Amount she has to spend = Price of the television + sales tax.]
100x+6.5x100 ≤ 788.10
106.5x100 ≤ 788.10 [Simplify.]
x ≤ 788.10×100106.5
x ≤ $740
So, the maximum price of the television Katie can afford to buy is $740.
3ax + 8b < - 6b [Subtracting 2ax from the two sides of the equation.]
3ax < -14b [Subtracting 8b from the two sides of the equation.]
Given a < 0, hence it is a negative value. The inequality sign will change when dividing by 3a.
x > - 14b3a
So, the solution set is {x: x > - 14b3a }
Correct answer : (1)
8.
Determine whether $a$ - $b$ > 0 when $a$ > $b$.
a.
No
b.
Yes
Solution:
a - b > 0 when a > b is true. Because, the addition or subtraction of any real number does not affect the inequality.
a > b
a - b > 0 [Subtracting b from the two sides of the inequality.]
Correct answer : (2)
9.
Tim has $25. He wants to buy some pens and each pen costs 73 cents. Find the maximum number of pens that Tim can buy.
a.
35
b.
32
c.
33
d.
34
Solution:
Let x be the number of pens Tim can buy.
Total cost of x pens = $73 / 100x [73 cents = $73 / 100.]
The amount with Tim is only $25. So, the total price of the pens shall not exceed $25.
73 / 100x ≤ 25 [Express as a relation.]
73x ≤ 2500 [Multiply throughout by 100.]
x ≤ 34.24657
So, Tim can buy 34 pens.
Correct answer : (4)
10.
Gary has to take four Math tests to complete his seventh grade. On each of the tests he can score a maximum of 25. His scores on three tests are 20, 22 and 16. What should be his score in the fourth test, so that he gets a total score of at least 77 on all four tests?
a.
atmost 18
b.
atmost 19
c.
atleast 24
d.
atleast 19
Solution:
Let n be the score of Gary in his fourth test.
n + 20 + 22 + 16 ≥ 77
n + 58 ≥ 77 [Simplify.]
n + 58 - 58 ≥ 77 - 58 [Subtracting 58 from the two sides of the equation.]
n ≥ 19 [Simplify.]
Gary should get at least 19 points in his fourth test.