Solve the inequality (2(x−2))/(5) − (1(x+5))/(6)

Question

Solve the inequality (2(x−2))/(5) − (1(x+5))/(6) < 0 and simplify 7sqrt(5)sqrt(7) by rationalizing the denominator.

ScanSolve AI Tutor · Accepted Answer

Here is the solution for question 6 from the image.

6a) Solve the inequality $\frac{2(x-2)}{5} - \frac{1(x+5)}{6} < 0$

Find a common denominator for the fractions.
The least common multiple of 5 and 6 is 30.
Multiply the first fraction by $\frac{6}{6}$ and the second fraction by $\frac{5}{5}$ .
$\frac{6 \cdot 2(x-2)}{6 \cdot 5} - \frac{5 \cdot 1(x+5)}{5 \cdot 6} < 0$ $\frac{12(x-2)}{30} - \frac{5(x+5)}{30} < 0$
Combine the fractions over the common denominator.
$\frac{12(x-2) - 5(x+5)}{30} < 0$
Expand the terms in the numerator.
$\frac{12x - 24 - 5x - 25}{30} < 0$
Combine like terms in the numerator.
$\frac{(12x - 5x) + (-24 - 25)}{30} < 0$ $\frac{7x - 49}{30} < 0$
Multiply both sides by 30 to clear the denominator.
$7x - 49 < 0$
Add 49 to both sides.
$7x < 49$
Divide both sides by 7.
$x < \frac{49}{7}$ $x < 7$
The final answer for 6a) is $\boxed{x < 7}$ .

6b) By rationalizing the denominator, simplify $\frac{7\sqrt{5}}{\sqrt{7}}$
To rationalize the denominator, multiply the numerator and the denominator by $\sqrt{7}$ .
$\frac{7\sqrt{5}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}$
Multiply the numerators and the denominators.
$\frac{7\sqrt{5} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}}$ $\frac{7\sqrt{5 \cdot 7}}{7}$ $\frac{7\sqrt{35}}{7}$
Simplify the expression by canceling out the common factor of 7.
$\sqrt{35}$
The final answer for 6b) is $\boxed{\sqrt{35}}$ .

Solve the inequality (2(x-2))/(5) - (1(x+5))/(6) < 0 and simplify 7sqrt(5)sqrt(7) by rationalizing the denominator.