ultimate guide to linear inequalities
TRANSCRIPT
Ultimate Guide
To
Solving
Linear Inequalities
Solving Linear Inequalities
1.4 Sets, Inequalities, and Interval Notation
1.5 Intersections, Unions, and Compound Inequalities
1.6 Absolute-Value Equations and Inequalities
OBJECTIVES
1.4 Sets, Inequalities, and Interval Notation
a Determine whether a given number is a solution of aninequality.
b Write interval notation for the solution set or the graphof an inequality.
c Solve an inequality using the addition principle and themultiplication principle and then graph the inequality.
d Solve applied problems by translating to inequalities.
1.4 Sets, Inequalities, and Interval Notation
Inequality
An inequality is a sentence containing
1.4 Sets, Inequalities, and Interval Notation
Solution of an Inequality
Any replacement or value for the variable that makes an inequality true is called a solution of the inequality. The set of all solutions is called the solution set. When all the solutions of an inequality have been found, we say that we have solved the inequality.
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
aDetermine whether a given number is a solution of aninequality.
1 Determine whether the given number is a solution of the inequality.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
aDetermine whether a given number is a solution of aninequality.
1
We substitute 5 for x and get 5 + 3 < 6, or 8 < 6, a false sentence. Therefore, 5 is not a solution.
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
aDetermine whether a given number is a solution of aninequality.
3 Determine whether the given number is a solution of the inequality.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
aDetermine whether a given number is a solution of aninequality.
3
We substitute –3 for x and get or a true sentence. Therefore, –3 is a solution.
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
The graph of an inequality is a drawing that represents its solutions.
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
4 Graph on the number line.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
4
The solutions are all real numbers less than 4, so we shade all numbers less than 4 on the number line. To indicate that 4 is not a solution, we use a right parenthesis “)” at 4.
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
Write interval notation for the given set or graph.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
bWrite interval notation for the solution set or the graph of an inequality.
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
Two inequalities are equivalent if they have the same solution set.
1.4 Sets, Inequalities, and Interval Notation
The Addition Principle for Inequalities
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
10 Solve and graph.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
10
We used the addition principle to show that the inequalities x + 5 > 1 and x > –4 are equivalent. The solution set is and consists of an infinite number of solutions. We cannot possibly check them all.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
10Instead, we can perform a partial check by substituting one member of the solution set (here we use –1) into the original inequality:
EXAMPLE Solution
Since 4 > 1 is true, we have a partial check. The solution set is or The graph is as follows:
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
10
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
11 Solve and graph.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
11
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
11
The inequalities and have the same meaning and the same solutions. The solution set is or more commonly, Using interval notation, we write that the solution set is The graph is as follows:
1.4 Sets, Inequalities, and Interval Notation
The Multiplication Principle for Inequalities
For any real numbers a and b, and any positive number c:
For any real numbers a and b, and any negative number c:
Similar statements hold for
1.4 Sets, Inequalities, and Interval Notation
The multiplication principle tells us that when we multiply or divide on both sides of an inequality by a negative number, we must reverse the inequality symbol to obtain an equivalent inequality.
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
13 Solve and graph.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
13
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
13
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
15 Solve.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
15
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
cSolve an inequality using the addition principle and themultiplication principle and then graph the inequality.
15
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
1.4 Sets, Inequalities, and Interval Notation
Translating “At Least” and “At Most”
EXAMPLE
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16 Cost of Higher Education.
The equation C = 126t + 1293 can be used to estimate the average cost of tuition and fees at two-year public institutions of higher education, where t is the number of years after 2000. Determine, in terms of an inequality, the years for which the cost will be more than $3000.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16
1. Familiarize. We already have a formula. To become more familiar with it, we might make a substitution for t. Suppose we want to know the cost 15 yr after 2000, or in 2015. We substitute 15 for t:
C = 126(15) + 1293 = $3183.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16
We see that in 2015, the cost of tuition and fees at two-year public institutions will be more than $3000. To find all the years in which the cost exceeds $3000, we could make other guesses less than 15, but it is more efficient to proceed to the next step.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16
2. Translate. The cost C is to be more than $3000. Thus we have C > 3000. We replace C with 126t + 1293 to find the values of t that are solutions of the inequality:
126t + 1293 > 3000.
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16
3. Solve. We solve the inequality:
EXAMPLE Solution
1.4 Sets, Inequalities, and Interval Notation
d Solve applied problems by translating to inequalities.
16
4. Check. A partial check is to substitute a value for t greater than 13.55. We did that in the Familiarize step and found that the cost was more than $3000.
5. State. The average cost of tuition and fees at two-year public institutions of higher education will be more than $3000 for years more than 13.55 yr after 2000, so we have