Question

solve the inequality and graph it's solution on a number line : -1/4< 1/2 - x/3 < 2 , x€ i​

Answer 1

Answer:

• x = { -4, -3, -2, -1, 0, 1, 2 }

or

• -4.5 < x < 2.25, x ∈ I

Step-by-step explanation:

Given inequality is,

$\sf\implies - \dfrac{1}{4} < \dfrac{1}{2} - \dfrac{x}{3} < 2, x \in I$

Subtract 1/2 from inequality.

${\sf\implies - \dfrac{1}{4} - \dfrac{1}{2} < \dfrac{1}{2} - \dfrac{x}{3} - \dfrac{1}{2} < 2 - \dfrac{1}{2} }$

${\sf\implies \dfrac{ - 1 - 2}{4}< - \dfrac{x}{3}< \dfrac{4 - 1}{2} }$

${\sf\implies \dfrac{ -3}{4}< - \dfrac{x}{3}< \dfrac{3}{2} }$

Multiply the inequality with 3.

${\sf\implies \dfrac{ -3}{4} \times 3< - \dfrac{x}{3} \times 3< \dfrac{3}{2} \times 3 }$

${\sf\implies \dfrac{ -9}{4} < -x< \dfrac{9}{2} }$

Multiply the inequality with -1

${\sf\implies \dfrac{ -9}{4} \times ( - 1) > -x \times (- 1) > \dfrac{9}{2} \times ( - 1) }$

${\sf\implies \dfrac{9}{4} > x > - \dfrac{ 9}{2} }$

${\sf\implies 2.25 > x > - 4.5}$

Now the integers lying between this interval are:

• { -4, -3, -2, -1, 0, 1, 2 }

[ I is the set of Integers ]

Refer to the number line in attachment.

$\rule{280}{1}$

Discover more:

Some properties of inequality:-

Addition or subtraction property:

• If we have x > y then x + z > y + z

We can add or subtract any real number on both sides of inequality, sign of inequality will remain unchanged.

Multiplication or division property:

• If we have x > y then xz > yz ( if z is +ve)
• If we have x > y then xz < yz ( if z is -ve)

Sign of inequality reverses when we multiply the both sides of inequality with a negative number and it remains same in case of positive number.

(Same goes for division also)

Reciprocal property:

• If we have x > y then 1/x < 1/y

Sign of inequality reverses in case when we reciprocate the terms present in the inequality.

Some common equality and inequality symbols:-

• < less than
• > greater than
• = equals to
• ≤ smaller than or equal to
• ≥ greater than or equal to
• ≈ approximately equal to
• ≠ not equal to

Answer 2

Answer:

I hope it helps

Step-by-step explanation:

the solution set is {-4,-3,-2,-1,0,1,2}