Chapter 7: Problem 9

By sketching an appropriate graph, or otherwise, solve the inequality $\frac{1}{2-x}<5$.

Short Answer

Expert verified

Answer: $(-∞,1.8)\cup(2,∞)$

Step by step solution

Rewrite the inequality

In order to simplify the inequality, we first subtract 5 from both sides to have all terms on one side. This gives us: $$\frac{1}{2-x}-5<0$$

Find a common denominator

In this step, we need to find a common denominator for the fractions on the left side of the inequality. The common denominator of $\frac{1}{2-x}$ and $5$ is $(2-x)$, so we multiply the second term by $\frac{2-x}{2-x}$ to obtain the common denominator: $$\frac{1}{2-x}-\frac{5(2-x)}{2-x}<0$$

Combine the fractions

Now that the fractions have the same denominator, we can combine them into one fraction: $$\frac{1-5(2-x)}{2-x}<0$$ Simplify the numerator: $$\frac{1-10+5x}{2-x}<0$$ $$\frac{5x-9}{2-x}<0$$

Identify the critical values

The critical values are the values of $x$ where the inequality is either equal to 0 or the fraction is undefined. For this inequality, the critical values are when the numerator is 0 $(5x-9=0)$ or when the denominator is 0 $(2-x=0)$. Solve these equations to find the critical values: $$5x-9=0 \Rightarrow x=1.8$$ $$2-x=0 \Rightarrow x=2$$ So, the critical values are $x=1.8$ and $x=2$.

Test intervals around the critical values

We will now test intervals around the critical values to determine when the inequality is true or false. We will test values less than 1.8, values between 1.8 and 2, and values greater than 2. Since the inequality is strict (less than 0), we will not include the critical values in our solution. 1. Test $x<1.8$: Choose $x=1$: $$\frac{5(1)-9}{2-1}=\frac{-4}{1}<0$$ This interval is part of the solution. 2. Test $1.80$$ This interval is not part of the solution. 3. Test \(x>2$: Choose $x=3$: $$\frac{5(3)-9}{2-3}=\frac{6}{-1}<0$$ This interval is part of the solution.

Write the solution

From the intervals we tested, the inequality is true when $x<1.8$ and when $x>2$. Therefore, the solution to the inequality is: $$(-\infty,1.8)\cup(2,\infty)$$

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By sketching an appropriate graph, or otherwise, solve the inequality \(\frac{1}{2-x}<5\).

Short Answer

Step by step solution

Rewrite the inequality

Find a common denominator

Combine the fractions

Identify the critical values

Test intervals around the critical values

Write the solution

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