# Prove by contraposition that if x does not equal 5 and is irrational then 3x/x-5 is irrational. Question
Equations and inequalities Prove by contraposition that if x does not equal 5 and is irrational then $$\displaystyle{3}\frac{{x}}{{x}}-{5}$$ is irrational. 2021-03-19
Suppose that 3x/(x-5) is rational, that is, $$\displaystyle{3}\frac{{x}}{{{x}-{5}}}={q}$$,
for some rational q. Then $$\displaystyle{3}{x}={q}{\left({x}-{5}\right)}={q}{x}-{5}{q}\to{\left({3}-{q}\right)}{x}=-{5}{q}\to{x}=-{5}\frac{{q}}{{{3}-{q}}}$$
Since q is rational, then 3-q and -5q are also rational, and their quotient is also rational, meaning that x is rational.
By contraposition, we have proved the required statement.

### Relevant Questions Fill in the bla
so the resulting statement is true.
when solving
$$3x^2+2y^2=35$$
$$4x^2+3y^2=48$$
by the addition method, we can eliminate $$x^2$$ by the multiplying the first equation by -4 and the second equation by __________ and then adding the equations If $$\displaystyle{x}^{{{2}}}={y}^{{{3}}}$$ and y>1, what does $$\displaystyle{x}^{{\frac{{{2}}}{{{3}}}}}$$ equal in terms of y? Determine which of the following are linear inequalities and linear equations.Write LI if the sentence is linear inequality and LE if it is linear equation
1.2x+y>1
2.y=3x-6
3.y-5<7x
4.$$3s >= t+1$$
5.x+5y=10 This is the quesetion. Suppose that a does not equal 0.
a. if $$\displaystyle{a}\cdot{b}={a}\cdot{c}$$, does it follow that b=c?
b. if $$\displaystyle{a}\times{b}={a}\times{c}$$, does it follow that b=c ?
c. if $$\displaystyle{a}\cdot{b}={a}\cdot{c}$$ and $$\displaystyle{a}\times{b}={a}\times{c}$$, does it follow that b=c?
Either prove the assertion is true in general or show that it is false for a concret choice of vectors a, b, c Fill in the bla
so the resulting statement is true
when solving
4x,-,3y=15
3x-2y=10
by the addition method we can eliminate y by multiplying the first equation by 2 and the second equation by ______, and then adding the equations $$\displaystyle{3}{x},−{3},−{3}{x}^{{{2}}}−{3}{x}^{{{2}}},−{5}{x}−{5}{x}$$
Which of these terms does not belong in the group and why? Prove that if $$\displaystyle{\left|{\left|{u}\right|}\right|}={\left|{\left|{v}\right|}\right|},{t}{h}{e}{n}{\left({u}+{v}\right)}·{\left({u}−{v}\right)}={0}.$$  Suppose a,b\inZ. If a∣b, then $$\displaystyle{a}^{{{2}}}∣{b}^{{{2}}}.$$ 