If we have |x| = 3 then the question is, “what number can x be so that the distance to zero is 3?” There are two solutions to this question -3 and 3.
In general,
Here n is a positive integer and X represents an algebraic expression called the argument of the absolute value. To solve these equations, set the argument equal to plus or minus n and solve the resulting equations.
Solve.
This technique requires us to first isolate the absolute value. Apply the usual steps for solving to obtain the absolute value alone on one side of the equation, and then set the argument to plus or minus n.
Solve.
When two absolute values are equal we can set the argument of one equal to the argument of the other.
Tip: When in doubt of a solution, check to see if it solves the original equation. For example, check that {-7, 3} is the solution set for,
Notice that both numbers solve the original equation and therefore we have verified they are in the solution set.
All of the steps that we have learned for solving linear equations are the same for solving linear inequalities except one. We may add or subtract any real number to both sides of an inequality and we may multiply or divide both sides by any positive real number.
The only new rule comes from multiplying or dividing by a negative number.
So whenever we divide or multiply by a negative number we must reverse the inequality. It is easy to forget to do this so take special care to watch out for negative coefficients.
Notice that we obtain infinitely many solutions for these linear inequalities. Because of this we have to present our solution set in some way other than a big list. The two most common ways to express solutions to an inequality are by graphing them on a number line and interval notation.
Note: We use the following symbol to denote infinity:
Tip: Always use round parentheses and open dots for inequalities without the equal and always use square brackets and closed dots for inequalities with the equal.
When setting up proportions, be sure to be consistent. Units for the numerators should be the same and units for the denominators should be the same as well. After obtaining an equation with two equal fractions, cross multiply.
Proportion Problem: If 2 out of 3 dentists prefer Crest toothpaste, how many prefer Crest out of 600 dentists surveyed?
Proportion Problem: In Visalia 3 out of every 7 voters said yes to proposition 40. If 42,000 people voted, how many said no to proposition 40?
Proportion Problem: A recipe calls for 5 tablespoons of sugar for every 8 cups of flour. How many tablespoons of sugar are required for 32 cups of flour?
Coin Problem: Sally has 12 coins consisting of quarters and dimes. The value adds to $2.25. How many of each coin does she have?
Mixture Problem: A 50% alcohol solution is mixed with a 10% alcohol solution to create 8 ounces of a 32% alcohol solution. How much of each is needed?
Interest Problem: Mary invested her total savings of $3,400 in two accounts. Her mutual fund account earned 8% interest last year and her CD earned 5%. If her total interest for the year was $245, how much was in each account?
Notice that both numbers solve the original equation and therefore we have verified they are in the solution set.
