EQUATIONS AND INEQUALITIES • Algebraic equations are mathematical equations that contain a letter or variable, which represents a number. • To solve an algebraic equation, inverse operations are used. Inverse operations are performing the opposite operation to what is being performed in the equation. The inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Multiplication and division are the inverse operations of each other. • Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; a nd less than or equal to, ≤. Inequalities are also solved by using inverse operations. • Often a number line is used to show inequalities. To show an inequality on the number line, a shaded circle is used to represent that the answer is equal to a number. A non-shaded circle means the answer is not equal to a number. A shaded arrow on the number line to the left of a circle means that the answer is less than the circled number. If the shaded arrow is to the right, the answer is greater than the circled number. How to use equations and inequalities: • Algebraic equations are solved using the inverse operations. To solve the equation n - 52 = 38, look to see what operation is being performed in the problem, subtraction. The inverse operation of subtraction is addition. The problem can be solved using the inverse operation of addition as the example shows: n - 52 = 38 + 52 +52 n = 90 Since 52 was subtracted in the initial problem, it will be added with inverse operations. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
• The rule when dealing with equations is whatever is done to one side of the equation must be done to the other side. So 52 is added to 38 on the other side of the equal sign. When both sides are added, the numbers on the left side of the equation will cancel out, leaving only the n. On the right side of the equation, 38 and 52 are added, giving the answer of 90. Therefore n = 90. • Inequalities are solved the same way as algebraic equations by using inverse operations. The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Ex. x + 3 ≥ 7 -2x < 28 - 3 -3 -2 -2 x ≥ 4 x > -14 (Notice sign is reversed) The result is x ≥ 4. This means that x can be any number that meets this requirement such as 4, 8, 10, 20 etc. • On the number line the answer would look as follows: • The inequality x < 2 on the number line would look as follows: • To write an inequality, translate the words into numbers and signs: greater than > greater than or equal to ≥ less than < less than or equal to ≤ © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Try This! Solve the following equations: x - 4 = 16 56 + x = 134 6x = 84 x/7 = 15 Solve and graph the following inequalities: x + 14 ≤ 35 x - 12 > 42 4x < 128 x/-2 ≥ 18 Write the following inequalities: © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.