Question

Solve each rational inequality. Give the solution set in interval notation. 4/(3+x)≤3/(3+x)

Accepted Answer

Start by writing down the inequality: $$\frac{4}{3+x} \leq \frac{3}{3+x}. $$Identify the domain restrictions by finding values that make the denominator zero. Set $$3 + x = 0$$ which gives $$x = -3. So$$, $$x 
eq -3$$ because the expression is undefined there. Since the denominators on both sides are the same and nonzero (except at $$x = -3$$), multiply both sides of the inequality by $$(3+x)^2 ($$which is always positive except at $$x = -3) to $$eliminate the denominators safely without reversing the inequality sign. This gives: $$4(3+x) \leq 3(3+x). $$Simplify the inequality: $$4(3+x) \leq 3(3+x)$$ becomes $$12 + 4x \leq 9 + 3x. $$Then, subtract $$9$$ and $$3x$$ from both sides to isolate $$x$$: $$12 - 9 + 4x - 3x \leq 0$$ which simplifies to $$3 + x \leq 0. $$Solve the simplified inequality $$3 + x \leq 0 to $$get $$x \leq -3. $$Remember to exclude $$x = -3$$ from the solution set because it makes the original expression undefined. Finally, express the solution in interval notation considering the domain restriction.

Solve each rational inequality. Give the solution set in interval notation. 4/(3+x)≤3/(3+x)

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Key Concepts

Rational Inequalities

Domain Restrictions

Solving Inequalities by Multiplying or Subtracting