Answer :
To solve the given inequality [tex]\(\sqrt{4x - 4} < \sqrt{x} - 1 + 3\)[/tex], let's follow these steps carefully:
1. Simplify the inequality:
The inequality can be rewritten as:
[tex]\[ \sqrt{4x - 4} < \sqrt{x} + 2 \][/tex]
2. Square both sides to eliminate the square roots:
However, before squaring both sides, it's important to determine the conditions necessary for the inequality to hold. Both sides of the inequality must be non-negative (since we are dealing with square roots). Therefore:
[tex]\[ 4x - 4 \geq 0 \implies x \geq 1 \][/tex]
Similarly,
[tex]\[ \sqrt{x} + 2 \geq 0 \implies x \geq 0 \][/tex]
Since [tex]\(x\)[/tex] must satisfy both conditions, we have [tex]\(x \geq 1\)[/tex].
Now, square both sides of the simplified inequality:
[tex]\[ (\sqrt{4x - 4})^2 < (\sqrt{x} + 2)^2 \][/tex]
This yields:
[tex]\[ 4x - 4 < x + 4\sqrt{x} + 4 \][/tex]
3. Simplify the resulting inequality:
[tex]\[ 4x - 4 < x + 4\sqrt{x} + 4 \][/tex]
Isolate all terms involving [tex]\(x\)[/tex] on one side:
[tex]\[ 4x - x - 8 < 4\sqrt{x} \][/tex]
[tex]\[ 3x - 8 < 4\sqrt{x} \][/tex]
4. Square both sides again to eliminate the square root:
[tex]\[ (3x - 8)^2 < (4\sqrt{x})^2 \][/tex]
This simplifies to:
[tex]\[ 9x^2 - 48x + 64 < 16x \][/tex]
Bring all terms to one side to form a quadratic inequality:
[tex]\[ 9x^2 - 64x + 64 < 0 \][/tex]
5. Solve the quadratic inequality:
We need to find the roots of the quadratic equation [tex]\(9x^2 - 64x + 64 = 0\)[/tex] and determine the intervals where the quadratic expression is less than zero.
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
[tex]\[ x = \frac{64 \pm \sqrt{64^2 - 4 \cdot 9 \cdot 64}}{2 \cdot 9} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{4096 - 2304}}{18} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{1792}}{18} \][/tex]
[tex]\[ x = \frac{64 \pm 8\sqrt{28}}{18} \][/tex]
[tex]\[ x = \frac{32 \pm 4\sqrt{7}}{9} \][/tex]
The roots of the quadratic equation are:
[tex]\[ x_1 = \frac{32 - 4\sqrt{7}}{9}, \quad x_2 = \frac{32 + 4\sqrt{7}}{9} \][/tex]
Evaluate these approximate values:
[tex]\[ x_1 \approx 1.37, \quad x_2 \approx 5.23 \][/tex]
The quadratic inequality [tex]\(9x^2 - 64x + 64 < 0\)[/tex] holds between the roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
6. Find the integral values in the interval:
The interval where the inequality holds true is approximately:
[tex]\[ 1.37 < x < 5.23 \][/tex]
The integer values in this interval are [tex]\(x = 2, 3, 4, 5\)[/tex].
Therefore, the number of integral values of [tex]\(x\)[/tex] that satisfy the inequality is:
[tex]\[ \boxed{4} \][/tex]
1. Simplify the inequality:
The inequality can be rewritten as:
[tex]\[ \sqrt{4x - 4} < \sqrt{x} + 2 \][/tex]
2. Square both sides to eliminate the square roots:
However, before squaring both sides, it's important to determine the conditions necessary for the inequality to hold. Both sides of the inequality must be non-negative (since we are dealing with square roots). Therefore:
[tex]\[ 4x - 4 \geq 0 \implies x \geq 1 \][/tex]
Similarly,
[tex]\[ \sqrt{x} + 2 \geq 0 \implies x \geq 0 \][/tex]
Since [tex]\(x\)[/tex] must satisfy both conditions, we have [tex]\(x \geq 1\)[/tex].
Now, square both sides of the simplified inequality:
[tex]\[ (\sqrt{4x - 4})^2 < (\sqrt{x} + 2)^2 \][/tex]
This yields:
[tex]\[ 4x - 4 < x + 4\sqrt{x} + 4 \][/tex]
3. Simplify the resulting inequality:
[tex]\[ 4x - 4 < x + 4\sqrt{x} + 4 \][/tex]
Isolate all terms involving [tex]\(x\)[/tex] on one side:
[tex]\[ 4x - x - 8 < 4\sqrt{x} \][/tex]
[tex]\[ 3x - 8 < 4\sqrt{x} \][/tex]
4. Square both sides again to eliminate the square root:
[tex]\[ (3x - 8)^2 < (4\sqrt{x})^2 \][/tex]
This simplifies to:
[tex]\[ 9x^2 - 48x + 64 < 16x \][/tex]
Bring all terms to one side to form a quadratic inequality:
[tex]\[ 9x^2 - 64x + 64 < 0 \][/tex]
5. Solve the quadratic inequality:
We need to find the roots of the quadratic equation [tex]\(9x^2 - 64x + 64 = 0\)[/tex] and determine the intervals where the quadratic expression is less than zero.
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
[tex]\[ x = \frac{64 \pm \sqrt{64^2 - 4 \cdot 9 \cdot 64}}{2 \cdot 9} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{4096 - 2304}}{18} \][/tex]
[tex]\[ x = \frac{64 \pm \sqrt{1792}}{18} \][/tex]
[tex]\[ x = \frac{64 \pm 8\sqrt{28}}{18} \][/tex]
[tex]\[ x = \frac{32 \pm 4\sqrt{7}}{9} \][/tex]
The roots of the quadratic equation are:
[tex]\[ x_1 = \frac{32 - 4\sqrt{7}}{9}, \quad x_2 = \frac{32 + 4\sqrt{7}}{9} \][/tex]
Evaluate these approximate values:
[tex]\[ x_1 \approx 1.37, \quad x_2 \approx 5.23 \][/tex]
The quadratic inequality [tex]\(9x^2 - 64x + 64 < 0\)[/tex] holds between the roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
6. Find the integral values in the interval:
The interval where the inequality holds true is approximately:
[tex]\[ 1.37 < x < 5.23 \][/tex]
The integer values in this interval are [tex]\(x = 2, 3, 4, 5\)[/tex].
Therefore, the number of integral values of [tex]\(x\)[/tex] that satisfy the inequality is:
[tex]\[ \boxed{4} \][/tex]
