Equations and inequalities

Linear equations

Linear equations are the most common task type in entrance tests. Harder versions put x in a denominator, so you first state which values are not allowed (the domain) and then clear the fractions.

München Named for the T, M and W maths tests at Studienkolleg München. Source

The rules

• Do the same to both sides: add, subtract, multiply or divide by the same non-zero number.
• Collect x terms on one side and numbers on the other, then divide by the coefficient.
• With x in a denominator: exclude values that make it zero, then multiply both sides by the common denominator.
• Check the solution by substituting it back.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Solve for x: (x − 8)/2 − (x + 1)/5 = -3

   • a-4
   • b3
   • c5
   • d4
   Show solution

   Answer: 4

   Multiply every term by the common denominator; mind the minus in front of the second fraction.

   Check: (4 − 8)/2 = -2 and (4 + 1)/5 = 1, and -2 − 1 = -3.

2. 2. Solve for x: 2(x − 1) + 2 = x + 2

   • a3
   • b1
   • c2
   • d-2
   Show solution

   Answer: 2

   Expand, collect x-terms on one side, constants on the other, divide.

   Expand: 2x − 2 + 2 = x + 2, then isolate x.

3. 3. Solve for x: 2(x − 1) + 4 = x − 1

   • a-3
   • b-4
   • c-2
   • d3
   Show solution

   Answer: -3

   Expand, collect x-terms on one side, constants on the other, divide.

   Expand: 2x − 2 + 4 = x − 1, then isolate x.

4. 4. Solve for x: 2(x + 3) − 7 = x − 3

   • a-3
   • b-2
   • c-1
   • d2
   Show solution

   Answer: -2

   Expand, collect x-terms on one side, constants on the other, divide.

   Expand: 2x + 6 − 7 = x − 3, then isolate x.

5. 5. Solve for x: 3(x − 1) + 1 = 2x − 6

   • a-3
   • b-4
   • c4
   • d-5
   Show solution

   Answer: -4

   Expand, collect x-terms on one side, constants on the other, divide.

   Expand: 3x − 3 + 1 = 2x − 6, then isolate x.

6. 6. Solve for x: 3(x − 4) + 24 = 2x + 6

   • a-5
   • b6
   • c-7
   • d-6
   Show solution

   Answer: -6

   Expand, collect x-terms on one side, constants on the other, divide.

   Expand: 3x − 12 + 24 = 2x + 6, then isolate x.

Common mistakes

• Multiplying only one term by the denominator instead of every term.
• Forgetting to exclude values from the domain.
• Sign errors when moving terms across the equals sign.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs