Linear Inequalities Test -7

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Linear Inequalities Test -7

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Question 1
1 / -0

If a , b , c are real numbers such that a < b , c ≥ 0, then

A
ac ≤ bc

B
ac > bc

C
ac < bc

D
none of these

Solution

The inequality remains same if both sides of an inequality are multiplied by the same positive real number and inequality will become an equation if we multiply it by 0.
Question 2
1 / -0

Region represented by the system x ≥ 0 , y ≥ 0 of inequations is

A
2^nd quadrant

B
1^st quadrant

C
3^rdquadrant

D
none of these.

Solution

The solution region of x ≥ 0 will be the half of XY –plane which lies on the right of y – axis , including the points on y – axis [ First and Fourth quadrants ]

The solution region of y ≥ 0 will be the half of XY –plane which lies above x – axis , including the points on x – axis [ First and Second quadrants ]

Hence the solution region of x ≥ 0 , y ≥ 0 will be the the intersection of the above two regions ,which is the first quadrant.
Question 3
1 / -0

If a , b , c are real numbers such that a ≤ b , c > 0, then

A
ac > bc

B
ac ≤ bc

C
ac ≥ bc

D
ac < bc

Solution

The inequality remains same if both sides of an inequality are multiplied by the same positive real number
Question 4
1 / -0

Solution set of the inequality y < 0 is

A
half of XOY – plane which lies below x -axis

B
half of XOY – plane which lies above x -axis

C
half of XOY – plane which lies below x –axis , including the points on x – axis

D
none of these

Solution

The solution region of y ≤ 0 will be the half of XY –plane which lies under x – axis , including the points on x – axis [ First and Second quadrants ]

Hence the solution region of y < 0 will be the half of XY –plane which lies below x – axis .

Since equality is not there it will not include any point on x – axis .
Question 5
1 / -0

If a , b , c are real numbers such that a ≤ b , c < 0 then

A
ac ≤ bc

B
ac > bc

C
ac ≥ bc

D
none of these

Solution

The sign of the inequality is to be reversed (≤ to ≥ or ≥ to ≤) if both sides of an inequality are multiplied by the same negative real number.
Question 6
1 / -0

If a , b , c are real numbers such that a > b , c < 0

A
ac > bc

B
ac < bc

C
ac ≥ bc

D
none of these

Solution

The sign of the inequality is to be reversed (< to > or > to <) if both sides of an inequality are multiplied by the same negative real number.
Question 7
1 / -0

If x < 5, then

A
− x < 5

B
x > − 5

C
− x > − 5

D
none of these .

Solution

Given x < 5

Multiplying both sides of the above inequality by -1,we get

− x > − 5 [The sign of the inequality is to be reversed if both sides of an inequality are multiplied by the same negative real number]
Question 8
1 / -0

Given that x , y and b are real numbers and x < y , b < 0, then

A
x/b ≥ y/b

B
x/b ≤ y/b

C
x/b < y/b

D
x/b > y/b

Solution

The sign of the inequality is to be reversed (< to > or > to <) if both sides of an inequality are divided by the same negative real number.
Question 9
1 / -0

Which of the following number line represents the solution of the inequality
$$15 < 4x + 3 \le 31$$ ?

A

B

C

D

Solution

$$15 < 4x + 3 \le 31$$ ........... (Given)
$$\implies 15-3 < 4x+3-3 \leq 31-3$$
$$\implies 12 < 4x \leq 28$$
$$\implies 3 < x \le 7$$ ......... (Divided by $$4$$)
Hence, option C is correct.
Question 10
1 / -0

Ordered pair that satisfy the equation $$x + y + 1 < 0$$ is:

A
$$(0, -1)$$

B
$$(-2 , 0)$$

C
$$(2, - 4)$$

D
Both (B) and (C)

Solution

Given inequation is $$x+y+1<0$$

From option A, $$0+\left ( -1 \right )+1 <0 $$
$$ \Rightarrow 0<0 $$ which is false
Hence, $$(0,-1)$$ is not a solution.

From option B, $$-2+0+1 <0 $$
$$ \Rightarrow -1<0 $$ which is true
Hence, $$(-2,0)$$ there is a solution

From option C, $$2-4+1 <0 $$
$$ \Rightarrow -1<0 $$ which is true.
Hence, $$(2,-4)$$ is a solution
Hence, Option B and C are the solutions.
Question 11
1 / -0

Which equation has the solution shown on the number line?

A
$$-3\leq x < 0 $$.

B
$$3\leq x < 6 $$.

C
$$x\neq 1$$

D
$$x < 0$$

Solution

The red line represents all the numbers that satisfy the equation $$-3\leq x < 0 $$.
Question 12
1 / -0

If you multiply an inequality by a negative number, when should you reverse the inequality symbol?

A
Always

B
Never

C
Sometimes

D
Only if the negative number is a fraction

Solution

If we take $$x\leq y$$ then $$-x\geq -y$$ always.
Thus, the inequality gets reversed always when multiplied by negative number.
Hence, option A is correct.
Question 13
1 / -0

Which equation has the solution shown on the number line?

A
$$x \geq 1$$

B
$$x \geq -6$$

C
$$x\neq 1$$

D
$$x < 0$$

Solution

The red line represents all the numbers that satisfy the equation $$x \geq -6$$.
Question 14
1 / -0

Which equation has the solution shown on the number line?

A
$$1\leq x < 4$$.

B
$$x < 1$$

C
$$x\neq 1$$

D
$$x < 0$$

Solution

The red line represents all the numbers that satisfy the equation $$1\leq x < 4$$.
Question 15
1 / -0

Find solution of following inequality also show it graphically.
$$x<5,x\in W$$.

A

B

C

D

Solution

Given, $$x<5$$ and $$x \in W$$
Natural numbers are counting numbers whose set is $$W=\{0,1,2,3,...\}$$
Therefore,$$\{0,1,2,3,4\}$$ represents $$x<5$$
Option A graph has $$\{0,1,2,3,4\}$$ solution set.
Question 16
1 / -0

Solve the following inequality and show it graphically:
$$-2<x+3<5,x\in Z$$

A

B

C

D

Solution

Given, $$-2<x+3<5$$
Subtracting $$3$$ from all sides, we get
$$-2-3<x+3-3<5-3$$
$$-5<x<2$$
Thus $$x$$ will contain all the points between $$-5$$ and $$2$$ except point $$-5$$ and $$2$$.
Question 17
1 / -0

Which of the following number line represents the solution of the inequality $$2x+1 \ge 9$$?

A

B

C

D

Solution

Given, $$ 2x+1 \geq 9$$
$$\Rightarrow 2x \geq 9-1$$
$$\Rightarrow 2x\geq 8$$
$$\Rightarrow x \geq 4$$
$$\therefore$$Solution $$=$$ $$x \geq 4$$
Hence, option $$A$$ is the answer.
Question 18
1 / -0

Find the solution of following inequality, also show it graphically:
$$x<4,x\in R$$

A

B

C

D

Solution

$$x<4$$ and $$x\in R$$
means x takes all real values less than 4.(but not 4)
So It is interval $$(-\infty,4)$$
The graph is option C.