Write and Solve Inequalities Chapter Notes | Mathematics for Grade 6 PDF Download

Introduction

Inequalities and equations are ways to compare or balance numbers in math. An inequality, like x < 15, shows one value is less than, greater than, or equal to another, and you can graph it on a number line to show all possible solutions. An equation, like x + 2 = 8, says two expressions are equal, and you solve it to find the value of the variable. Using inverse operations, like adding or dividing, helps find solutions. This guide explains inequalities and one-step equations with simple examples.

What is an inequality?

• An inequality is a comparison statement. Here are two examples of inequalities:
  • x is less than 15
    x < 15
  • y ≥ –9
    y is greater than or equal to -9
• The table below shows symbols you might see in an inequality:
• A solution to an inequality is a number that makes the inequality true. For example, 8 is a solution to the inequality x < 15 because 8 < 15.

How do you graph an inequality on a number line?

• You can use a number line to show all possible solutions to an inequality.
• For Examples:
  Let's try it with y ≥ – 9.
  Start by graphing y = –9. Since there is an equal sign, put a dot at – 9.
  Now, draw an arrow pointing to the right to show numbers greater than – 9.
  So, this graph shows y ≥ – 9.
  Now, let's graph x < 15.
  Since x = 15 is not a solution, draw an open circle at 15. Then, draw an arrow pointing to the left to show numbers less than 15.
  So, this graph shows x < 15.

Solving equations

• An equation is a mathematical sentence that says two expressions are equal. Some equations have a variable. A variable is a letter that represents an unknown amount. Here are some equations with variables:
  • k + 9 = 32
  • 15 – y = 3
  • 270 = 9n
  • t/6 = 17
• To find a solution of an equation with a variable, find a value for the variable that makes the equation true.
• For example, the solution to the equation 2 + x = 8 is x = 6. Replacing x with 6 makes the equation true.
  • 2 + x = 8
  • 2 + 6 = 8
• You can often find the solution to a one-variable equation by getting the variable alone on one side of the equation. You can use inverse operations to help. 

One-step equations with addition and subtraction

• If you have an equation with addition or subtraction, you can apply the inverse operation to both sides of the equation to get the variable alone. Addition and subtraction are inverse operations.
• For Example: Solve k – 19 = 51. The equation has subtraction. Use the inverse operation, addition.
  k–19 = 51
  k – 19 + 19 = 51 + 19 ... Add 19 to both sides of the equation.
  k = 70    Simplify.
  You can check your solution. Take the original equation, and replace the variable with your solution. If that equation is true, your solution is correct!
  70 – 19 = 51
  51 = 51  This is true, so k = 70 is the solution.

One-step equations with multiplication and division

• If you have an equation with multiplication or division, you can apply the inverse operation to both sides of the equation to get the variable alone. Multiplication and division are inverse operations.
• For Example: Solve 12m = 156. The equation has multiplication. Use the inverse operation, division.
  12m = 156
  12m/12 = 156/12 Divide both sides of the equation by 12.
  m = 13 Simplify.
  Then, check your solution.
  12 x 13 = 156
  156 = 156  This is true, so m = 13 is the solution.