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# Algebraic Inequalities Topic

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ALGEBRAIC INEQUALITIES Algebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. • Inequalities can be solved using addition, subtraction, multiplication and division. • One-step inequalities and two-step inequalities are solved by using inverse operations. Most two-step inequalities involve either addition or subtraction with either multiplication or division. • With inequalities, attention must be given when multiplying or dividing by a negative number. When this occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. • When inequalities are written with words or word problems, the words must be changed into the correct numbers, variables and signs in order to determine the correct answer. • 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 the number. A non-shaded circle means the answer is not equal to the 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. • Some algebraic inequalities can contain variables on both sides of the inequality. In this case, the variables and numbers need to be moved so there are only variables on one side of the inequality sign and numbers on the other side of the inequality sign before evaluating. © 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.

How to use algebraic inequalities • Algebraic inequalities written as words or word problems must be changed into the correct numbers, variables and sign before solving. • When writing an inequality, translate the words into signs as follows: greater than > greater than or equal to ≥ less than < less than or equal to ≤ For example, what is the phrase two times a number decreased by four is less then fifty-two as a mathematical inequality? Ex. Two times a number → 2x decreased by four → -4 is less than fifty-two → < 52 The equation is 2x - 4 < 52. • Once the words or word problems are changed, the inequality can be evaluated. To evaluate two-step inequalities, inverse operations are used. With two-step inequalities, it is very important to isolate the variable before evaluating. Isolating the variable means to get the variable alone on one side of the inequality. • 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. © 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.

For example, evaluate -4x + 6 ≥ 34. Ex. -4x + 6 ≥ 34 → isolate the variable by subtracting 6 - 6 -6 -4x ≥ 28 → now solve for x by dividing -4 -4 -4 x ≤ 7 ← notice that the sign is reversed In this inequality, x ≤ 7. If 4 were divided first and then 6 was subtracted, the result would be incorrect. The correct result is x ≤ 7, which means that x can be any number that meets this requirement such, as -12, -5, 0, 2, 7 etc. On the number line the answer would look as follows: • Algebraic inequalities can also have variables on both sides of an inequality sign. To solve for these inequalities, the variables must be on one side of the inequality sign and the numbers must be on the other side of the inequality sign. Then the inequality can be evaluated. Try This! What is the algebraic inequality that means a number multiplied by six minus five is greater than or equal to nineteen? Solve for x, 8x - 10 < 46 Solve for x, x/4 + 12 ≥ 14 Solve for x, 7 - 3x ≤ 43 Solve for x, x/-5 - 9 > - 5 Solve for x, x + 4 ≤ 2x – 3 Solve for x, 3x - 6 > x - 8 © 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.