Inequalities — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Solve the inequality for x.

   $$x+3<8$$

   Answer: $x<5$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x<5$$
2. 2.

   Solve the inequality for x.

   $$2x\le 10$$

   Answer: $x\le 5$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x\le 5$$
3. 3.

   Solve the inequality for x.

   $$-x>4$$

   Answer: $x<-4$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x<-4$$
4. 4.

   Solve the inequality for x.

   $$3x-2\ge 7$$

   Answer: $x\ge 3$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x\ge 3$$
5. 5.

   Solve the inequality for x.

   $$5-2x<9$$

   Answer: $x>-2$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x>-2$$
6. 6.

   Solve the inequality for x.

   $$3-3x\le 12$$

   Answer: $x\ge -3$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x\ge -3$$
7. 7.

   Solve the inequality for x.

   $$x+5<9$$

   Answer: $x<4$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x<4$$
8. 8.

   Solve the inequality for x.

   $$3x\le 12$$

   Answer: $x\le 4$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x\le 4$$
9. 9.

   Solve the inequality for x.

   $$-x>7$$

   Answer: $x<-7$

   Step-by-step solution

   1. $$\text{Keep both sides ordered while isolating }x.$$
   2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
   3. $$x<-7$$
10. 10.

    Solve the inequality for x.

    $$2x-6\ge 4$$

    Answer: $x\ge 5$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x\ge 5$$
11. 11.

    Solve the inequality for x.

    $$8-2x<2$$

    Answer: $x>3$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x>3$$
12. 12.

    Solve the inequality for x.

    $$-2x-2\le 6$$

    Answer: $x\ge -4$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x\ge -4$$
13. 13.

    Solve the inequality for x.

    $$x-2>6$$

    Answer: $x>8$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x>8$$
14. 14.

    Solve the inequality for x.

    $$4x<20$$

    Answer: $x<5$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x<5$$
15. 15.

    Solve the inequality for x.

    $$-5x\ge 15$$

    Answer: $x\le -3$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x\le -3$$
16. 16.

    Solve the inequality for x.

    $$3-x\le 1$$

    Answer: $x\ge 2$

    Step-by-step solution

    1. $$\text{Keep both sides ordered while isolating }x.$$
    2. $$\text{Reverse the inequality only when multiplying or dividing by a negative.}$$
    3. $$x\ge 2$$

More from Math Foundations

• Order of Operations
• Signed Numbers
• Fraction Sense
• Ratios and Percentages
• Exponents and Roots
• Algebraic Expressions
• Linear Equations
• Polynomial Basics
• Coordinate Plane
• Reading Graphs
• Function Machines
• Substitution
• Factoring Basics