2.4 – Linear inequalities and problem solving

Objectives I will use interval notation. I will solve linear inequalities using the addition property of inequality. I will solve linear inequalities using the multiplication property of inequality. I will solve problems that can be modeled by linear inequalities.

Why are linear inequalities important? Suppose a salesperson earns a base salary of $600 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total of at least $1500 per month. Here, the phrase “at least” implies that an income of $1500 or more is acceptable. In symbols, we can write… income ≥ 1500

Linear Inequality in one variable A linear inequality in one variable is an inequality that can be written in the form ax + b < c Where a, b, and c are real numbers and a ≠ 0

Solution A solution of an inequality is a value of the variable that makes the inequality a true statement.

Solution Set The solution set of an inequality is the set of all solutions. x > 2 Contains all numbers greater than 2 The graph of {x/ x>2} looks like the following. 2

Interval Notation With interval notation use a parenthesis instead of an open circle. The graph of {x/ x>2} will now look like this… This graph can be represented in interval notation as (2, ∞) The graph parenthesis indicates that 2 is NOT included in the interval. 2

Interval Notation Use a bracket if a number IS included in the interval. To graph, {x / x ≥ 2} use a bracket instead of a parenthesis. This inequality written in set notation is [2, ∞) 2

Check your understanding Set Notation: {x / x < a} Graph: Interval Notation: (-∞, a) See page 84 a

Example 1 Graph each set on a number line and then write in interval notation. a. {x/ x ≥ 2} b. {x / x < -1} c. {x/ 0.5 <x ≤3

Addition Property of Inequality If a, b, and c are real numbers, then a < b and a + c < b + c

Example 2 x – 2 < 5 x – 2 + 2 < 5 + 2 x < 7 Solve x -2 < 5. Graph the solution set. x – 2 < 5 x – 2 + 2 < 5 + 2 x < 7 The solution set is {x / x < 7} Interval notation is (- ∞, 7) 7 8 6

You try it! Write the solution in set notation, interval notation, and graph Solve: x +3 < 1

Solution Set: {x / x ≥ -10} Interval Notation: [-10, ∞) Example 3 Solve 3x + 4 ≥ 2 x – 6 3 x + 4 ≥ 2 x – 6 3 x + 4 – 4 ≥ 2x – 6 – 4 3x ≥ 2x – 10 3x – 2x ≥ 2x – 10 – 2x x ≥ -10 Solution Set: {x / x ≥ -10} Interval Notation: [-10, ∞) [ -10 -9 -11

You try it! Solve: 5 x – 1 ≥ 4 x + 4

Multiplication property of inequality If a, b, and c are real numbers and c is positive then a < b and ac < bc are equivalent inequalities. If a,b,c and c are real numbers and c is negative, then a < b and ac > bc

Hint Whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol must be reversed to form an equivalent inequality.

Example 4: Solve and graph the solution set. Multiply both sides by 4 Inequality symbol is the same since we are multiplying by a positive number.

Example 4: Solve and graph the solution set. -2.3 -2.3 x > -3 Set Notation {x/ x > -3} Interval Notation (-3, ∞) Inequality symbol is reversed since we divided by a negative number.

-2x > 7 2x – 3 > 10 -x + 4 + 3x < 7 - x + 4 < 5 Concept Check - In which of the following inequalities must the inequality symbol be reversed during the solution? -2x > 7 2x – 3 > 10 -x + 4 + 3x < 7 - x + 4 < 5

Steps for solving a linear inequality in one variable Step 1: Clear the equation of fractions Step 2: Use distributive property to remove grouping symbols. Step 3: Combine like terms on each side of the inequality. Step 4: Use the addition property of inequality Step 5: Use the multiplication property of inequality. ULTIMATE GOAL- ISOLATE THE VARIABLE!

Example 5 Solve: - (x – 3) + 2 ≤ 3 (2x – 5) + x -x + 5 + x ≤ 7 x – 15 + x 20 ≤ 8 x Write the problem Distributive property Combine like terms Add x to both sides Add 15 to both sides Divide both sides by 8 and simplify

You try it!  -(2x – 6) ≤ 4 (2x – 4) + 2

Example 6: Solve: Write inequality Clear the fractions Distribute Subtract 5 x from both sides Add 12 to both sides Divide both sides by negative 3, this will reverse inequality symbol!

You try it! 

Example 7: Solve 2 (x+3) > 2x + 1 2x + 6 – 2x > 2x + 1 – 2x 6 > 1 6 >1 is a true statement for all values of x, so this inequality and the original inequality are true for all numbers. {x / x is a real number} OR (-∞ , ∞)

You try it!  5(x – 3) < 5x + 2

Example 8 – Calculating income with commission A salesperson earns $600 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total income of at least $1500 per month. TRANSLATE: Let x = amount of sales 600 + 0.20x ≥ 1500

Interpret: The income for sales of $4500 is Solve 600 + 0.20x ≥ 1500 600 + 0.20x - 600 ≥ 1500 – 600 0.20x ≥ 900 x ≥ 4500 Interpret: The income for sales of $4500 is 600 + 0.20 (4500) = 1500 Thus, if sales are greater than or equal to $4500, income is greater than or equal to $1500.

Try it!  A salesperson earns $1000 a month plus a commission of 15% of sales. Find the minimum amount of sales needed to receive a total income of at least $4000 per month.

Example 9 – Finding annual consumption In the U.S., the annual consumption of cigarettes is declining. The consumption c in billions of cigarettes per year since the year 1985 can be approximated by the formula c = -14.25t + 598.69 Where t is the number of years after 1985. Use this formula to predict the years that the consumption of cigarettes will be less than 200 billion per year.

Understand Read and re-read the problem. To become familiar with the given formula, let’s find the cigarette consumption after 25 years, which would be the year 1985 + 25, or 2010. To do so, we substitute 25 for t in the given formula. c = -14.25 (25) + 598.69 c = 242.44 Thus, in 2010, we predict cigarette consumption to be about 242.44 billion.

Translate We are looking for the years that consumption of cigarettes c is less than 200. Since we are finding years t, we substitute the expression in the formula for c, or -14.25 t + 598.69 < 200

Solve: -14.25 t + 598.69 < 200 -14.25 t + 598.69 < 200

Interpret: We substitute a number greater than 27.98 and see that c is less than 200. The annual consumption of cigarettes will be less than 200 billion for the years more than 27.98 years after 1985, or approximately 28 + 1985 = 2013.

You Try It!  Use the formula c = -14.25t + 598.69 to predict when the consumption of cigarettes will be less than 100 billion per year.