[3 marks sum]

Question 13 Marks

Given x ∈ {real numbers}, find the range of values of x for which -5 ≤ 2x – 3 < x + 2 and represent it on a number line.

Answer

-5 $\leq$ 2x - 3 < x + 2
-5 $\leq$ 2x - 3 and 2x - 3 < x + 2
-2 $\leq$ 2x and x < 5
-1 $\leq$ x and x < 5
Required rage is -1 $\leq$ x < 5
The required graph is:

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Question 23 Marks

Find the range of values of x which satisfies
$-2 \frac{2}{3} \leq x+\frac{1}{3}<3 \frac{1}{3} ; x \in R$
Graph these values of x on the number line.

Answer

$-2 \frac{2}{3} \leq x+\frac{1}{3} \text { and } x+\frac{1}{3}<3 \frac{1}{3} $
$ \Rightarrow-\frac{8}{3} \leq x+\frac{1}{3} \text { and } x+\frac{1}{3}<\frac{10}{3} $
$ \Rightarrow-\frac{8}{3} \leq x+\frac{1}{3} \text { and } x+\frac{1}{3}<\frac{10}{3} $
$ \Rightarrow-\frac{9}{3} \leq x \text { and } x<\frac{9}{3}$
=> -3 <= x and x < 3
$\therefore-3 \leq x<3$
The required graph of the solution set is:

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Question 33 Marks

x ∈ {real numbers} and -1 < 3 – 2x ≤ 7, evaluate x and represent it on a number line.

Answer

-1 < 3 - 2x ≤ 7
-1 < 3 - 2x and 3 - 2x ≤ 7
2x < 4 and - 2x ≤ 4
x < 2 and x ≥ - 2
Solution set = {-2 ≤ x < 2, x ∈ R}
Thus, the solution can be represented on a number line as:

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Question 43 Marks

Solve the following inequation, write the solution set and represent it on the number line.
$-3(x-7) \geq 15-7 x>\frac{x+1}{3}, x \in R$

Answer

$\Rightarrow 3(x-7) \geq 15-7 x>\frac{x+1}{3}, x \in R $
$ \Rightarrow-3(x-7) \geq 15-7 x \text { and } 15-7 x>\frac{x+1}{3} $$\Rightarrow-3 x+21 \geq 15-21 \text { and } 45-1>x+21 x $
$ \Rightarrow 4 x \geq-6 \text { and } 44>22 x $
$ \Rightarrow x \geq \frac{-3}{2} \text { and } 2>x $
$ \Rightarrow x \geq-1.5 \text { and } 2>x$
The solution set is $\{x: x \in R,-1.5 \leq x<2\}$
The solution set is represented on number line as follows:

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Question 53 Marks

Solve the following in equation and write the solution set:
13x - 5 < 15x + 4 < 7x + 12, x ∈ R
Represent the solution on a real number line.

Answer

$13 x-5<15 x+4<7 x+12, x \in R$
we have

Solution set $=\{x: x \in R$ and -4.5 < $x$ <1 }
The required line is

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Question 63 Marks

Solve the inequation:
3z – 5 ≤ z + 3 < 5z – 9, z ∈ R.
Graph the solution set on the number line

Answer

3z - 5 ≤ z + 3 < 5z - 9
3z - 5 ≤ z + 3 and z + 3 < 5z - 9
2z ≤ 8 and 12 < 4z
z ≤ 4 and 3 < z
Since, z ∈ R
∴ Solution set = {3 < z ≤ 4, x ∈ R }
It can be represented on a number line as:

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Question 73 Marks

Solve the given inequation and graph the solution on the number line.
2y – 3 < y + 1 ≤ 4y + 7, y ∈ R

Answer

2y – 3 < y + 1 ≤ 4y + 7, y ∈ R
⇒ 2y – 3 – y < y + 1 – y ≤ 4y + 7 – y
⇒ y – 3 < 1 ≤ 3y + 7
⇒ y – 3 < 1 and 1 ≤ 3y + 7
⇒ y < 4 and 3y ≥ 6 ⇒ y ≥ – 2
⇒ – 2 ≤ y < 4
The graph of the given equation can be represented on a number line as:

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Question 83 Marks

Find three consecutive largest positive integers such that the sum of one-third of first, one-fourth of second and one-fifth of third is almost $20.$

Answer

Let the required integers be $x, x + 1$ and $x + 2.$
According to the given statement,
$\frac{1}{3} x+\frac{1}{4}(x+1)+\frac{1}{5}(x+2) \leq 20 $
$ \frac{20 x+15 x+15+12 x+24}{60} \leq 20 $
$ 47 x+39 \leq 1200$
$47 \times \leq 1161 $
$x \leq 24.702$
Thus the largest values of the positive interger $x$ is $24,$
Hence the required integer are $24, 25$ and $26$

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Question 93 Marks

Solve the inequation
$-2 \frac{1}{2}+2 x \leq \frac{4 x}{5} \leq \frac{4}{3}+2 x, x \in W$
Graph the solution set on the number line.

Answer

$-2 \frac{1}{2}+2 x \leq \frac{4 x}{5} \leq \frac{4}{3}+2 x$
$ -2 \frac{1}{2} \leq \frac{4 x}{5}-2 x \leq \frac{4}{3} $
$ -\frac{5}{2} \leq-\frac{6 x}{5} \leq \frac{4}{3} $
$ \frac{25}{12} \geq x \geq-\frac{10}{9}$
$2.083 >= x >= -1.111$
Since $x \in W$
$\therefore$ Solution set $= {0, 1, 2}$
The solution set can be represented on number line as:

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Question 103 Marks

Find the set of values of x satisfying
$7 x+3 \geq 3 x-5$ and $\frac{x}{4}-5 \leq \frac{5}{4}-x$ where $x \in N$

Answer

$7 x+3 \geq 3 x-5 $
$ 4 x \geq-8$
$ x \geq-2 $
$ \frac{x}{4}-5 \leq \frac{5}{4}-x$
$\frac{x}{4}+x \leq \frac{5}{4}+5 $
$\frac{5 x}{4} \leq \frac{25}{4} $
$ x \leq 5$
Since $x \in N$
$\therefore$ Solution set $= \{1, 2, 3, 4, 5\}$

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Question 113 Marks

Given:
A = {x: 11x – 5 > 7x + 3, x ∈ R} and
B = {x: 18x – 9 ≥ 15 + 12x, x ∈ R}.
Find the range of set A ∩ B and represent it on the number line.

Answer

A = {x: 11x – 5 > 7x + 3, x ∈ R}
= {x: 4x > 8, x ∈ R}
= {x: x > 2, x ∈ R}
B = {x: 18x – 9 ≥ 15 + 12x, x ∈ R}
= {x: 6x ≥ 24, x ∈ R}
= {x: x ≥ 4, x ∈ R}
A ∩ B = {x: x ≥ 4, x ∈ R}
It can be represented on number line as:

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Question 123 Marks

Solve the following inequation and represent the solution set on the number line 2x – 5 ≤ 5x +4 < 11, where x ∈ I

Answer

2x - 5 $\leq$ 5x + 4 and 5x + 4 < 11
2x - 5x $\leq$ - 4 + 5 and 5x < 11 - 4
3x $\leq$ -9 and 5x < 7
$x \geq \frac{9}{-3}$ and $x<\frac{7}{5}$
$x \geq-3$ and $x<1 \frac{2}{5}$
-3 $\leq$ x $\leq$ $1\frac{2}{5}$
Since $x \in I$, the solution set is $\{-3,-2-1,0,1\}$.

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Question 133 Marks

Given: $A=\{x:-8<5 x+2 \leq 17, x \in I\}, B=\{x:-2 \leq 7+3 x<17, x \in R\}$
Where $R= \{real numbers\}$ and $I= \{integers\}.$ Represent $A$ and $B$ on two different number lines. Write down the elements of $A \cap B$.

Answer

$A=\{x:-8<5 x+2 \leq 17, x \in I$
$=\{x:-10<5 x<=15, x \in I\}$
$={x:-2}$ It can be represented on number line as

$B =\{ x :-2<=7+3 x <17, x \in R \}$
$=\{ x : 9<=3 x <10, x \in R \}$
$=\{x:-3 \leq x<3.33, x \in R\}$
it can be represeneted on number line as

$A \cap B=\{-1,0,1,2,3\}$

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Question 143 Marks

If $P =\{x: 7x — 4 > 5x + 2, x \in R\}$ and $Q = \{x: x — 19 \geq 1 — 3x, x \in R\},$ find the range of set $P \cap Q$ and represent it on a number line.

Answer

$P = \{x : 7x - 4 > 5x + 2, x \in R\}$
$7x - 4 > 5x + 2$
$7x - 5x > 2 + 4$
$2x > 6$
$x > 3$
$Q = \{x : x - 19 >= 1 - 3x, x \in R\}$
$x-19 \geq 1-3 x$
$x+3 x \geq 1+19$
$4 x \geq 20$
$x \geq 5$
$P \cap Q =\{x: x \geq 5, x \in R\}$

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Question 153 Marks

Use the real number line to find the range of values of x for which:-1 < x ≤ 6 and -2 ≤ x ≤ 3

Answer

-1 < x ≤ 6 and -2 ≤ x ≤ 3
Both the given inequations are true in the range where their graphs on the real number lines overlap.
The graphs of the given inequations can be drawn as:

From both graphs, it is clear that their common range is 1 < x ≤ 3

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Question 163 Marks

Use the real number line to find the range of values of x for which:

x < 0 and -3 ≤ x < 1

Answer

x < 0 and -3 ≤ x < 1
Both the given inequations are true in the range where their graphs on the real number lines overlap.
The graphs of the given inequations can be drawn as.

From both graphs, it is clear that their common range is -3 ≤ x < 0

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Question 173 Marks

Use the real number line to find the range of values of x for which:

x > 3 and 0 < x < 6

Answer

x > 3 and 0 < x < 6
Both the given inequations are true in the range where their graphs on the real number lines overlap.
The graphs of the given inequations can be drawn as:

From both graphs, it is clear that their common range is
3 < x < 6

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Question 183 Marks

The diagram represents two inequations A and B on real number lines:

1) Write down A and B in set builder notation
2) Represent A ∪ B and A ∩ B' on two different number lines

Answer

1) A = {x ∈ R: -2 ≤ x < 5}
B = {x ∈ R: -4 ≤ x < 3}
2) A ∩ B = {x ∈ R: -2 ≤ x < 5}
It can be represented on number line as:

B’ = {x ∈ R: 3 < x ≤ -4}
A ∩ B’ = {x ∈ R: 3 ≤ x < 5}
It can be represented on number line as:

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Question 193 Marks

Solve and graph the solution set of:

5 > p – 1 > 2 or 7 ≤ 2p – 1 ≤ 17, p ∈ R

Answer

5 > p – 1 > 2 or 7 ≤ 2p – 1 ≤ 17
6 > p > 3 or 8 ≤ 2p ≤ 18
6 > p > 3 or 4 ≤ p ≤ 9
Graph of solution set of 6 > p > 3 or 4 ≤ p ≤ 9
= Graph of points which belong to 6 > p > 3 or 4 ≤ p ≤ 9 or both
= Graph of points which belong to 3 < p ≤ 9
Thus, the graph of the solution set is:

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Question 203 Marks

Solve and graph the solution set of:
$3x – 2 > 19$ or $3 – 2x \leq -7, x \in R$

Answer

$3 x-2>19 \text { or } 3-2 x \geq-7 $
$ 3 x>21 \text { or }-2 x \geq-10 $
$ x>7 \text { or } x \leq 5$
Graph of solution set of $x > 7 or x \leq 5 =$ Graph of points which belong to $x > 7 or x \leq 5$ or both.
Thus, the graph of the solution set is:

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Mathematics [3 marks sum]

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