Why solve inequalities

This example presents a small problem. How can we indicate on the number line? If we estimate the point, then another person might misread the statement. Could you possibly tell if the point represents or maybe? Since the purpose of a graph is to clarify, always label the endpoint. A graph is used to communicate a statement. You should always name the zero point to show direction and also the endpoint or points to be exact. The solutions for inequalities generally involve the same basic rules as equations.

There is one exception, which we will soon discover. The first rule, however, is similar to that used in solving equations. If the same quantity is added to each side of an inequality , the results are unequal in the same order. Note that the procedure is the same as in solving equations. We will now use the addition rule to illustrate an important concept concerning multiplication or division of inequalities.

Remember, adding the same quantity to both sides of an inequality does not change its direction. To obtain x on the left side we must divide each term by - 2. Notice that since we are dividing by a negative number, we must change the direction of the inequality. Notice that as soon as we divide by a negative quantity, we must change the direction of the inequality.

Take special note of this fact. Each time you divide or multiply by a negative number, you must change the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities. When we multiply or divide by a positive number, there is no change.

When we multiply or divide by a negative number, the direction of the inequality changes. Be careful-this is the source of many errors. Once we have removed parentheses and have only individual terms in an expression, the procedure for finding a solution is almost like that in chapter 2.

Let us now review the step-by-step method from chapter 2 and note the difference when solving inequalities. First Eliminate fractions by multiplying all terms by the least common denominator of all fractions. No change when we are multiplying by a positive number.

Second Simplify by combining like terms on each side of the inequality. No change Third Add or subtract quantities to obtain the unknown on one side and the numbers on the other. No change Fourth Divide each term of the inequality by the coefficient of the unknown.

If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will be reversed. This is the important difference between equations and inequalities. The only possible difference is in the final step. What must be done when dividing by a negative number?

Solve equations and inequalities Simplify expressions Factor polynomials Graph equations and inequalities Advanced solvers All solvers Tutorials. Partial Fractions. Welcome to Quickmath Solvers! New Example. Help Tutorial. Solve an equation, inequality or a system. Equations and Inequalities Involving Signed Numbers In chapter 2 we established rules for solving equations using the numbers of arithmetic.

Thus we obtain Remember, abx is the same as 1abx. In this example we could multiply both numerator and denominator of the answer by - l this does not change the value of the answer and obtain The advantage of this last expression over the first is that there are not so many negative signs in the answer. And that works well for adding and subtracting , because if we add or subtract the same amount from both sides, it does not affect the inequality. Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

No matter, just swap sides, but reverse the sign so it still "points at" the correct value! Another thing we do is multiply or divide both sides by a value just as in Algebra - Multiplying. Do not try dividing by a variable to solve an inequality unless you know the variable is always positive, or always negative. Because we are multiplying by a negative number, the inequalities change direction. But to be neat it is better to have the smaller number on the left, larger on the right.

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