# Algebra Exercises - Equations and Inequalities

1) If a < b, then -a > -b Why?

2) What role of operations that applies when you are solving an equation does not apply when you are solving an inequality?

---Give examples to explain your answers.

Section 1.6: Exercises 100 and 104

Section 2.1: Exercises 92 and 94

Section 2.2: Exercises 78, 86, and 90

Section 2.3: Exercises 14, 22, 32, 54, and 68

Section 2.4: Exercises 32, 54, 72, and 82

Section 2.6: Exercise 8

Please see attached for the details of the problems.

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#### Solution Preview

Solutions:

1.) For this problem, it is best to explain with the aid of a number line. Suppose a < b, then b is much farther to zero and a to the right since b is much greater.

If using the same numbers we take the negative, we can see that b is still farther to a and zero but to the left.

Any number getting farther to zero to the left will have lesser value. So when we take the negative of a and b, the conditions will be reversed: -a > -b. For example, if a = 2 and b = 5 then: 2 < 5. But if we get the negative of each, -a = -2 and -b = -5, then: -2 > -5.

2.) There are certain operations in equality that you cannot simply apply in inequalities:

- When multiplying or dividing negative numbers on both sides, the sense of inequality must be reversed. A common mistake is the solver will forget to reverse the sense of inequality when multiplying or dividing a negative number. For example, -x > -10. When we multiply -1 to both sides, the inequality must be x < 10, and NOT x > 10.

- One cannot simply relate two inequalities having a different sense. Meaning, if a < b and b > c, one cannot ...

#### Solution Summary

This posting contains the solution and step-by-step procedures to solve each of the given problems.