SOLVING EXPONENTIAL EQUATIONS AND INEQUALITIES

Solve each inequality :

Problem 1 :

625 ≥ 5^{a + 8}

Solution :

625 ≥ 5^{a + 8}

Writing 625 in exponential form.

5⁴ ≥ 5^{a + 8}

4 ≥ a + 8

4 - 8 ≥ a

-4 ≥ a

a ≤ -4

So, the solution is a ≤ -4.

Problem 2 :

16 ^{2x -3} < 8

Solution :

16^{2x -3} < 8

16 = 2⁴ and 8 = 2³

2^{4(2x -3)} < 2³

Equating the powers, we get

4(2x - 3) < 3

2x - 3 < 3/4

2x < (3/4) + 3

2x < 15/4

x < 15/8

So, the solution is x < 15/8.

Problem 3 :

3^{2x - 1} ≥ (1/243)

Solution :

3^{2x - 1} ≥ (1/243)

243 = 3⁵

3^{2x - 1} ≥ (1/3⁵)

3^{2x - 1} ≥ 3^-5

Equating the powers, we get

2x - 1 ≥ -5

2x ≥ -5 + 1

2x ≥ - 4

x ≥ -4/2

x ≥ -2

So, the solution is x ≥ -2.

Problem 4 :

2^{x + 2} > (1/32)

Solution :

2^{x + 2} > (1/32)

32 = 2⁵

2^{x + 2} > (1/2⁵)

2^{x + 2} > 2^-5

Equating the powers, we get

x + 2 > -5

x > -5-2

x > -7

So, the solution is x > -7.

Problem 5 :

4^{2x + 6} ≤ 64^{2x - 4}

Solution :

4^{2x + 6} ≤ 64^{2x - 4}

64 = 4³

4^{2x + 6} ≤ 4^{3(2x - 4)}

Equating the powers, we get

2x + 6 ≤ 3(2x - 4)

2x + 6 ≤ 6x - 12

2x - 6x ≤ -12 - 6

-4x ≤ -18

Divide by -4, we get

x ≥ 18/4

x ≥ 9/2

Since we are dividing by the negative values, we have to flip the sign.

So, the solution is x ≥ 9/2.

Problem 6 :

25^{y - 3} ≤ (1/125)^y+2

Solution :

25^{y - 3} ≤ (1/125)^y+2

5^{2(y - 3)} ≤ (1/5³)^(y+2)

5^{2(y - 3)} ≤ 5^-3(y+2)

Equating the powers, we get

2(y - 3) ≤ -3(y + 2)

2y - 6 ≤ -3y - 6

2y + 3y ≤ -6 + 6

5y ≤ 0

Dividing by 5, we get

y ≤ 0/5

y ≥ 0

So, the solution is y ≥ 0.

Problem 7 :

Solution :

-2(3t + 5) ≥ -5(t - 6)

Distributing -2 and -5, we get

-6t - 10 ≥ -5t + 30

-6t + 5t ≥ 30 + 10

-t ≥ 40

t ≤ -40

Problem 8 :

Solution :

-2(w + 2) < -3(4w)

-2w - 4 < -12w

-2w + 12w < 4

10w < 4

w < 4/10

w < 2/5