Linear Inequalities Chapter 13

Recall: Equations 6x = 12 x = 2 x = 2 is the only solution There is only one solution!

Which fruit will tip the scale? Greater than

100 g 210 g 90 g 180 g 170 g 240 g The Fruits Greater than

The Outcomes Greater than

What did you observe? Any fruit which is greater than 100g will tip the scale. Weight of fruit > 100 Algebraically we can write this as x > 100 Let x be the weight of the fruit. Greater than

What did you observe? Let’s look at this on the number line. It’s easy to see that any fruit which lies to the right of 100 will tip the scale. The set of numbers that satisfy the inequality is called the solution set of the inequality. The solution set is represented on the number line like this. An empty circle is used to show that 100 is not included. Greater than There is more than 1 solution!

When will the balloon burst? This is a latex balloon in a vacuum chamber. As we move the slider to the left, the pressure in the chamber reduces. When the pressure in the chamber is less than the pressure inside the balloon, the balloon bursts. 180 190 200 210 220 230 The balloon burst when the pressure in the chamber is less than 204 mmHg Less than

When will the balloon burst? Pressure < 204 mmHg Algebraically, P < 204 On the number line, we have 180 190 200 210 220 230 204 Less than There is more than 1 solution! An empty circle is used to show that 204 is not included.

Bukit Timah The highest elevation in Singapore is Bukit Timah situated at 164m above sea level. Write an equality that describes any place in Singapore in comparison with Bukit Timah. The elevation at any place is less than or equal to 164m. If x represents the elevation at any place, x ≤ 164 Let’s represent this inequality on a number line. Less than or equal to The symbol ≤ is used to represent less than or equal to. A filled circle is used to show that 164 is included.

A filled circle is used to show that 19.4 is included. Singapore’s Weather The lowest recorded temperature in Singapore is 19.4°C. Write an inequality which describes the temperature in Singapore on any day in the past. The temperature on any day was greater than or equal to 19.4°C. If T represents the temperature, T ≥ 19.4 Let’s represent this inequality on a number line. Greater than or equal to The symbol ≥ is used to represent greater than or equal to.

Operations on Inequalities

Property 1 Let’s begin with equations first. When working with equations we know that If a = b then a + x = b + x and a – x = b – x This applies to inequalities as well. Adding or subtracting the same number on both sides of an equation leaves the equation unchanged. If a > b then ( a + x ) > ( b + x ) If a > b then ( a – x ) > ( b – x ) 4 > 1 ( 4 + 2 ) > ( 1 + 2 ) 7 > -3 ( 7 – 5 ) > ( -3 – 5 ) If a < b then ( a + x ) < ( b + x ) If a < b then ( a – x ) < ( b – x ) -2 < 8 ( -2 + 3 ) < ( 8 + 3 ) -5 < -2 ( -5 – 1 ) < ( -2 – 1 ) Adding or subtracting the same number on both sides of an inequality leaves the inequality unchanged.

For equations we know that ax = bx and a ÷ x = b ÷ x (x ≠ 0) Property 2 Multiplying or dividing by the same number on both sides of an equation leaves the equation unchanged. Does this rule apply for inequalities as well? 10 > 5 ( 10 × 2 ) > ( 5 × 2 ) -3 < 5 ( -3 × 3 ) < ( 5 × 3 ) -12 < -8 ( -12 ÷ 4 ) < ( -8 ÷ 4 ) 6 > -2 ( 6 ÷ 2 ) > ( -2 ÷ 2 ) Multiplying or dividing by the same positive number on both sides of the inequality leaves the inequality unchanged.

When x is a positive number, Property 2 If a > b then ( ax ) > ( bx ) If a > b then ( a ÷ x ) > ( b ÷ x ) If a < b then ( ax ) < ( bx ) If a < b then ( a ÷ x ) < ( b ÷ x ) Does this work when x is negative?

Now let’s see what happens when we multiply or divide each side of an inequality by the same negative number. Property 3 10 > 5 ( 10 × -2 ) > ( 5 × -2 ) ( 10 × -2 ) < ( 5 × -2 ) Notice that the inequality is reversed. Example 1

Now let’s see what happens when we multiply or divide each side of an inequality by the same negative number. Property 3 -2 < 4 ( -2 × -3 ) < ( 4 × -3 ) ( -2 × -3 ) > ( 4 × -3 ) Notice that the inequality is reversed. Example 2

Now let’s see what happens when we multiply or divide each side of an inequality by the same negative number. Property 3 6 > -2 ( 6 ÷ -2 ) > ( -2 ÷ -2 ) ( 6 ÷ -2 ) < ( -2 ÷ -2 ) Notice that the inequality is reversed. Example 3

Now let’s see what happens when we multiply or divide each side of an inequality by the same negative number. Property 3 -12 < -8 ( -12 ÷ -4 ) < ( -8 ÷ -4 ) ( -12 ÷ -4 ) > ( -8 ÷ -4 ) Notice that the inequality is reversed. Example 4

When x is a negative number, Property 3 If a > b then ( ax ) < ( bx ) If a > b then ( a ÷ x ) < ( b ÷ x ) If a < b then ( ax ) > ( bx ) If a < b then ( a ÷ x ) > ( b ÷ x ) Multiplying or dividing by the same negative number on both sides of an inequality reverses the inequality.

Solving Inequalities In the last two matches, a basketball player scored 22 and 35 goals. How many goals must she score in her next match is she is to have an average of at least 28 goals per match? Any number greater than 27 is ok. We can represent the solutions on a number line.

In the last two matches, a basketball player scored 22 and 35 goals. How many goals must she score in her next match is she is to have an average of at least 28 goals per match? The better way to solve this problem! Let x be the third score in her next match. Average = ( 22 + 35 + x ) ÷ 3 But her average score should be at least 28 goals per match. So, ( 22 + 35 + x ) ÷ 3 ≥ 28 22 + 35 + x ≥ 3 × 28 22 + 35 + x ≥ 84 57 + x ≥ 84 x ≥ 84 – 57 x ≥ 27 Notice that 27 is included. Multiply both sides by 3 Subtract both sides by 57 I must score at least 27 goals!

Example 1 Solve the inequality 4x < 28. 4x < 28 x < 28 ÷ 4 x < 7 Try it 1! Solve the inequality 3x < 36. 3x < 36 x < 36 ÷ 3 x < 12

Example 2 Solve the inequality . Try it 2! Solve the inequality .

Homework 13.1 Do on foolscap paper. Q 1, 2, 3, 4, 5, 6 Not compulsory to draw number lines. But it is good to draw for word problems.

Example 3 The maximum load of a lift is 360 kg. If we assume that the mass of each boy is 40 kg, find the possible number of boys who can use the lift at any one time. Let x be the number of boys. 40x ≤ 360 x ≤ 360 ÷ 40 x ≤ 9 The possible number of boys who can use the lift are 1, 2, 3, 4 ,5, 6, 7, 8 and 9. Try it 3! The maximum load a small trolley can carry is 40 kg. If the mass of each carton of goods is 5 kg, find the possible number of cartons the trolley can carry at any one time. Let x be the number of cartons. 5x ≤ 40 x ≤ 40 ÷ 5 x ≤ 8 The possible number of cartons the trolley can carry are 1, 2, 3, 4 ,5, 6, 7 and 8.

Example 4 One ham and egg sandwich contains 22g of protein. Suppose the minimum daily intake of a man is 48g of protein. Find the least whole number of ham and egg sandwich that the man should eat should eat per day to meet his minimum daily protein requirement. Let x be the number of sandwiches the man is required to eat in a day. 22x ≥ 48 x ≥ 48 ÷ 22 x ≥ The man should eat at least 3 sandwiches a day. Try it 4! A bowl of congee contains 70g of carbohydrate. Suppose the minimum daily requirement of a man is 130g, find the minimum whole number of bowls of congee that the man should eat per day to meet the minimum carbohydrate requirement. Let x be the number of bowls of congee the man is required to eat in a day. 70x ≥ 130 x ≥ 130 ÷ 70 x ≥ The man should at least eat 2 bowls a day.

Homework 13.2 Do on foolscap paper. Q 1, 2, 3, 4, 5, 6, 7, 8 Not compulsory to draw number lines. But it is good to draw for word problems.