Calculus 1 Exercise : Inequalities (Question and Answers)

CALCULUS I

INEQUALITIES

By Ponta

1. Determine the solution set of the inequality 𝟕𝒙 − 𝟐 ≤ 𝟗𝒙 + 𝟑.
Ans:

𝟕𝒙 − 𝟐 + (−𝟑) ≤ 𝟗𝒙 + 𝟑 + (−𝟑) (Add −𝟑 to both sides)

𝟕𝒙 − 𝟓 ≤ 𝟗𝒙

(−𝟕𝒙) + 𝟕𝒙 − 𝟓 ≤ (−𝟕𝒙) + 𝟗𝒙 (Add −𝟕𝒙 to both sides)

−𝟓 ≤ 𝟐𝒙

𝟐𝒙 ≥ −𝟓
𝟏 𝟏
( ) × 𝟐𝒙 ≥ ( ) × (−𝟓) (Multiply both sides by ½)
𝟐 𝟐
𝟓
𝒙 ≥ (− )
𝟐
The solution set that satisfies the inequality is
𝟓
❖ {𝒙|𝒙 ≥ − 𝟐}


2. Determine the solution set of the inequality 𝟏. 𝟗𝟗 < 𝟏/𝒙 < 𝟐. 𝟎𝟏
Ans:
(1) From Left
𝟏. 𝟗𝟗 < 𝟏/𝒙
𝟏
𝟏. 𝟗𝟗 × (𝒙) < × (𝒙) (Multiply both sides by 𝒙)
𝒙
𝟏. 𝟗𝟗𝒙 < 𝟏
𝟏
𝒙 < 𝟏/(𝟏. 𝟗𝟗) (Multiply both sides by 𝟏.𝟗𝟗)

(2) From Right
𝟏/𝒙 < 𝟐. 𝟎𝟏
𝟏 < 𝟐. 𝟎𝟏𝒙 (Multiply both sides by 𝒙)