Question

sqrt(5x - 4) + sqrt(3x + 1) < 3​

Answer 1

x ∈(1,40) when the expression is $\sqrt{5x-4}+\sqrt{3x+1} < 3$ by taking square on both sides of the expression.

Given that,

$\sqrt{5x-4}+\sqrt{3x+1} < 3$

We have to solve the square root function and find the value of x.

We know that,

Take the expression

$\sqrt{5x-4}+\sqrt{3x+1} < 3$

Now, checking what is the square root value of each term

First, 5x - 4 ≥ 0

x ≥ $\frac{4}{5}$

And 3x + 1 ≥0

x ≥ $\frac{-1}{3}$

So,

$\sqrt{5x-4}+\sqrt{3x+1} < 3$

$\sqrt{5x-4} < 3-\sqrt{3x+1}$

Taking square on both sides,

5x - 4 < 9 + 3x + 1 - 6$\sqrt{3x+1}$

5x -3x -4 -10 < -6$\sqrt{3x+1}$

2x - 14 < -6$\sqrt{3x+1}$

x - 7 < -3$\sqrt{3x+1}$

Taking square on both sides,

(x - 7)² < (-3$\sqrt{3x+1}$)²

x² - 14x + 49 < 9 (3x+1)

x² -14x + 49 < 27x + 9

x² -14x -27x + 49 -9 <0

x² - 41x + 40 < 0

(x - 1)(x - 40) < 0

x ∈(1,40)

Therefore, x ∈(1,40) when the expression is $\sqrt{5x-4}+\sqrt{3x+1} < 3$

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Answer 2

EXPLANATION.

⇒ √(5x - 4) + √(3x + 1) < 3.

We need to find the domain of the expression.

if any expression written in square root the,

square root must be ≥ 0.

Now, we can write expression as,

⇒ 5x - 4 ≥ 0.

⇒ 5x ≥ 4.

⇒ x ≥ 4/5. - - - - - (1).

⇒ 3x + 1 ≥ 0.

⇒ 3x ≥ - 1.

⇒ x ≥ - 1/3. - - - - - (2).

Now, we shift √(3x + 1) in RHS, we get.

⇒ √(5x - 4) < 3 - √(3x + 1).

Squaring on both sides of the expression, we get.

⇒ [√(5x - 4)]² < [3 - √(3x + 1)]².

⇒ (5x - 4) < (3)² + [√(3x + 1)]² - 2(3)(√(3x + 1).

⇒ 5x - 4 < 9 + 3x + 1 - 6√(3x + 1).

⇒ 5x - 4 < 10 + 3x - 6√(3x + 1).

⇒ 5x - 4 - 10 - 3x < - 6√(3x + 1).

⇒ 2x - 14 < - 6√(3x + 1).

⇒ 2(x - 7) < - 6√(3x + 1).

⇒ (x - 7) < - 3√(3x + 1).

Now, again squaring on both sides of the expression, we get.

⇒ (x - 7)² < [-3√(3x + 1)]².

⇒ x² + 49 - 14x < 9(3x + 1).

⇒ x² + 49 - 14x < 27x + 9.

⇒ x² + 49 - 14x - 27x - 9 < 0.

⇒ x² - 41x + 40 < 0.

Factorizes the equation into middle term splits, we get.

⇒ x² - 40x - x + 40 < 0.

⇒ x(x - 40) - 1(x - 40) < 0.

⇒ (x - 1)(x - 40) < 0.

Now, we find the zeroes of the expression, we get.

⇒ (x - 1)(x - 40) = 0.

⇒ x = 1  and  x = 40.

Put this point on wavy curve method, we get.

⇒ x ∈ (1, 40).

∴ √(5x - 4) + √(3x + 1) < 3 is equal to x ∈ (1, 40).

                                                                                                                   

MORE INFORMATION.

Rules of inequality.

We can multiply Or divide any non - zero number "k" on both sides of inequality sign of inequality will change according to sign of k that is,

If k > 0 then sign of inequality will remain same.

If k < 0 then sign of inequality will get reversed.

Wavy curve method.

Step - 1 : Check structure.

Step - 2 : Find critical points.

Step - 3 : Plot all critical points on number lines.

Step - 4 : Check sign of given example by putting value of x from different parts of number line.