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Chapter 1.7 Inequalities

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An inequality says that one expression is greater than, or greater than or equal to, or less than, or less than or equal to, another. As with equations, a value of the variable for which the inequality is true is a solution of the inequality. Two inequalities with the same solution set are equivalent.

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Properties of Inequality For real numbers a, b, and c: If a<b, then a + c < b + c, If a 0 then ac < bc, If a bc.

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Properties of Inequality Multiplication may be replaced by division in properties 2 and 3. Always remember to reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

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Linear Inequalities in One Variable A linear inequality in one variable is an inequality that can be written in the form ax + b > 0

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Solve -3x + 5 > -7 Example 1 Solving a Linear Inequality

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The original inequality is satisfied by any real number less than 4. The solution set can be written {x| x < 4}. A graph of the solution set is shown in Figure 9, where the parenthesis is used to show that 4 itself does not belong to the solution set.

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Open Interval (a(a

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(a(a )b)b

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)b)b

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Half-open Interval [a[a

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Half-opened Interval (a(a ]b]b

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[a[a )b)b

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Half-open Interval ]b]b

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Closed Interval [a[a ]b]b

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All real numbers

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Example 2 Solving a Linear Inequality Give the solution set in interval notation and graph it.

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Interval Notation [ 054321-5-4-3-2

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Three-Part Inequalities The inequality -2 < 5 +3x < 20 is the next example that 5 + 3x is between -2 and 20. This inequality is solved using an extension of the properties of inequality given earlier, working with all three expressions at the same time.

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Example 3 Solving a Three-Part Inequality Solve -2 < 5 +3x < 20

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Graph 054321-5-4-3-2

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A product will break even, or begin to produce a profit, only if the revenue from selling the product at least equals the cost of producing it. If R represents revenue and C is cost, then the break- even point is the point where R = C.

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If the revenue and cost of a certain product are given by R = 4x and C = 2x + 1000 where x is the number of units produced and sold, at what production level does R at least equal C? Example 4 Finding the Break-Eve Point

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If the revenue and cost of a certain product are given by R = 4x and C = 2x + 1000 where x is the number of units produced and sold, at what production level does R at least equal C? Example 4 Finding the Break-Eve Point

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Quadratic Inequality A quadratic inequality is an inequality that can be written in the form ax 2 + bx + c < 0

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Example 5 Solving a Quadratic Inequality Solve x 2 – x - 12

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y x

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Example 6 Solving a Quadratic Inequality

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y x

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If a projectile is launched from ground with an initial velocity of 96 ft per sec. its height in feet t seconds after launching is s feet, where s = -16t 2 + 96 t When will the projectile be greater than 80 ft above ground level. Example 7 Solving a Problem Involving the Height of a Projectile

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Rational Inequalities Inequalities involving rational expressions such as are called rational inequalities, and are solved in a manner similar to the procedure for solving quadratic inequalities.

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Example 8 Solving a Rational Inequality

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y x

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Example 9 Solving a Rational Inequality

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y x

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y x

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Homework Section 1.7 # 1 - 52

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