 # Inequalities

In mathematics, the inequality relation is one of the fundamental building blocks for inequalities. It deals with the mathematical comparison of two elements, and it serves as a key to understanding why numbers behave the way they do. Inequalities allow us to describe sets of numbers and the relationship between those numbers.

##### 2019 NJC P1 Q2

Without using a calculator

(i)

Solve the inequality $\frac{2{{x}^{2}}+5x+9}{x-7}\le x-4$.



(i) Solve the inequality $\frac{2{{x}^{2}}+5x+9}{x-7}\le x-4$.



(ii)

Hence, solve the inequality $\frac{2+5{{e}^{x}}+9{{e}^{2x}}}{7{{e}^{x}}-1}\ge 4{{e}^{x}}-1$.



(ii) Hence, solve the inequality $\frac{2+5{{e}^{x}}+9{{e}^{2x}}}{7{{e}^{x}}-1}\ge 4{{e}^{x}}-1$.



##### Suggested Handwritten and Video Solutions      ##### 2013 CJC Promo Q8

(a)

(i)

Without using a calculator, solve the inequality $\frac{x+6}{{{x}^{2}}-3x-4}\le \frac{1}{4-x}$.



(i) Without using a calculator, solve the inequality $\frac{x+6}{{{x}^{2}}-3x-4}\le \frac{1}{4-x}$.



(ii)

Hence, deduce the range of values of $x$ that satisfies

$\frac{\left| x \right|+6}{{{x}^{2}}-3\left| x \right|-4}\le \frac{1}{4-\left| x \right|}$.



(ii) Hence, deduce the range of values of $x$ that satisfies

$\frac{\left| x \right|+6}{{{x}^{2}}-3\left| x \right|-4}\le \frac{1}{4-\left| x \right|}$.



(b)

Solve the inequality $\ln \left( x+6 \right)\le -\frac{x}{3}$.



(b) Solve the inequality $\ln \left( x+6 \right)\le -\frac{x}{3}$.



##### Suggested Handwritten and Video Solutions          ##### 2020 CJC Promo Q1

Without using a calculator, solve

$x<\frac{3}{x-2}$.



Hence solve

${{\text{e}}^{-x}}<\frac{3}{{{\text{e}}^{-x}}-2}$.



##### Suggested Handwritten and Video Solutions  ##### 2019 TJC Promo Q1

(i)

Sketch the curve with equation $y=\left| \frac{\alpha x}{x+1} \right|$, where $\alpha$ is a positive constant, stating the equations of the asymptotes. On the same diagram, sketch the line with equation $y=\alpha x-2$.



(i) Sketch the curve with equation $y=\left| \frac{\alpha x}{x+1} \right|$, where $\alpha$ is a positive constant, stating the equations of the asymptotes. On the same diagram, sketch the line with equation $y=\alpha x-2$.



(ii)

Solve the inequality $\left| \frac{\alpha x}{x+1} \right|\ge \alpha x-2$, giving your answers in term of $\alpha$.



(ii) Solve the inequality $\left| \frac{\alpha x}{x+1} \right|\ge \alpha x-2$, giving your answers in term of $\alpha$.



##### Suggested Handwritten and Video Solutions      ##### 2013 ACJC P1 Q1

Do not use a graphic calculator in answering this question.
By considering the cases $x<0$ and $x\ge 0$, find the range of values of $x$ that satisfy the inequality

$\left( 10-x \right)\left( 10-\left| x \right| \right)>11$,



##### Suggested Handwritten and Video Solutions    Play Video

H2 Math Question Bank Check out our question bank, where our students have access to thousands of H2 Math questions with video and handwritten solutions.  