Graphing Linear Inequalities

Graphing Linear Inequalities6.1 & 6.2

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Check whether the given number is a solution of the Inequality.

2. 4m > –23; –6

ANSWER yes

ANSWER no

1. x3

≤ 7; 20

Warm-Up

Add.

5) 6)

ANSWER yes

ANSWER–4

3. Is 4 a solution of the inequality 2a – 1.5 ≥ 6?

4. Solve the equation of + 8 = 7.f4

253

21

322

34

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

What’s the Difference?• 4x + 2 = 3y • 4x + 2 < 3y

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Steps to Solving Linear Inequalities• Solve for the variable.

▫ If you multiple or divide by a negative number, then you change the sign of the inequality < changes to > > changes to < Example: -2x < 5

▫ Otherwise, solve as you normally would (if their was an equal sign).

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Graph the Linear Inequality• Graph it on a number

line.

▫ The number is not a solution (> or <)

▫ The number is a solution (≤ or ≥)

• x > -6.5

• x ≤ 4

a.

b.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

x – 5 > – 3.5

x – 5 + 5 > – 3.5 + 5

x > 1.5

ANSWER

The solutions are all real numbers greater than 1.5. Check by substituting a number greater than 1.5 for x in the original inequality.

x – 5 > – 3.5. Graph your solution.Solve

Write original inequality.

Add 5 to each side.

Simplify.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

Solve the inequality. Graph your solution.

x – 9 ≤ 3

x – 9 + 9 ≤ 3 + 9

x ≤ 12

Write original inequality.

Add 9 to each side.

Simplify.

4. x – 9 ≤ 3

SOLUTION

ANSWER The solutions are all real numbers less than 12. Check by substituting a number less than 12 for x in the original inequality.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

SOLUTION

5. p – 9.2 < – 5

p – 9.2 < – 5

p – 9.2 + 9.2 < – 5 + 9.2

x < 4.2

Write original inequality.

Add 9.2 to each side.

Simplify.

ANSWER The solutions are all real numbers less than 4.2. Check by substituting a number less than 4.2 for x in the original inequality.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

Write original inequality.

Simplify.

6. – 1 ≥ m – 12

– 1 ≥ m – 12

– 1 + ≥ m +12

12

12

≥ m12

–

Add to each side.12

ANSWER The solutions are all real numbers less than . Check by substituting a number less than for x in the original inequality.1

2–

12

–

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

EXAMPLE 4

Solve 9 x + 7. Graph your solution.>9 ≥ x + 7 Write original inequality.

9 – 7 ≥ x + 7 – 7

Subtract 7 from each side.

2 ≥ x Simplify.

ANSWER

You can rewrite 2 ≥ x as x ≥ 2. The solutions are all real numbers less than or equal to 2.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

EXAMPLE 1

x 4

< 5. Write original inequality.x

4<4 4 5 Multiply each side by 4.

x < 20 Simplify.

ANSWER

The solutions are all real numbers less than 20. Check by substituting a number less than 20 in the original inequality.

. Graph your solution.x

4< 5Solve

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Solve the inequality. Graph your solution.

1. > 8x3

x 3

> 8. Write original inequality.

Multiply each side by 8.

x > 24 Simplify.

ANSWER The solutions are all real numbers are greater than 24. Check by substituting a number greater than 24 in the original inequality.

SOLUTION

x 3

> 83 3

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

2. ≤ – 2m8

Write original inequality.

Multiply each side by 8.

Simplify.

ANSWER The solutions are all real numbers less than are equal to – 16. Check by substituting a number less than – 16 in the original inequality.

SOLUTION

m 8

≤ – 2

m < – 16

m 8

8 3≤ – 2

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

Write original inequality.

Multiply each side by 2.5.

Simplify.

ANSWER The solutions are all real numbers less than are equal to – 10. Check by substituting a number greater than – 10 in the original inequality.

SOLUTION

y ≥ – 10

3. ≥ – 4y

2.5

≥ – 4y

2.5

8 2.5≥ – 4y

2.5

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

EXAMPLE 2

< 7.x–6

Write original inequality.

Multiply each side by –6. Reverseinequality symbol.

x > – 42 Simplify.

ANSWERThe solutions are all real numbers greater than – 42. Check by substituting a number greater than – 42 in the original inequality.

< 7. Graph your solution.x–6

Solve

x–6

> –6–6 7

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

EXAMPLE 2

CHECK

Write original inequality.

Substitute 0 for x.

Solution checks

x– 6

< 7

0– 6

< 7?

0 < 7

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

EXAMPLE 3

–3x > 24.–3x–3 < 24

–3

x < – 8

Write original inequality.

Divide each side by –3. Reverseinequality symbol.

Simplify.

–3x > 24.Solve

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

Multiply each side by – 4.

Simplify.

ANSWER

The solutions are all real numbers greater than are equal to – 48. Check by substituting a number greater than – 48 in the original inequality.

SOLUTION

x <– 48

4. > 12

x– 4

x– 4 > 12

– 4 12< – 4x– 4

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

Multiply each side by – 7.

Simplify.

SOLUTION

m < – 11.2

5. < 1.6

m– 7

m– 7 < 1.6

– 7 1.6< – 7m– 7

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

ANSWER

The solutions are all real numbers greater than are equal to 9. Check by substituting a number greater than 9 in the original inequality.

SOLUTION

6. 5v ≥ 45

5v ≥ 45

5v5 ≥ 45

5v ≥ 9

Divide both the side by 5.

Simplify.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.GUIDED PRACTICE

Write original inequality.

ANSWER

The solutions are all real numbers greater than are equal to – 4. Check by substituting a number greater than – 4 in the original inequality.

SOLUTION

7. – 6n < 24

– 6n < 24

>– 6n6

246

n > – 4

Divide both the side by 6.

Simplify.

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Solve a two-step inequality

EXAMPLE 1

3x – 7 < 8 Write original inequality.

3x < 15 Add 7 to each side.

x < 5 Divide each side by 3.

ANSWER

The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality.

3x – 7 < 8. Graph your solution.Solve

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Solve a two-step inequality

EXAMPLE 1

CHECK Write original inequality.

Solution checks.

Substitute 0 for x.?3(0) – 7 < 8

–7 < 8

3x –7 < 8

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Solve a multi-step inequality

EXAMPLE 2

Write original inequality.–0.6(x – 5) < 15 –

–0.6x + 3 < 15– Distributive property

Subtract 3 from each side.

– x >–20 Divide each side by 0.6. Reverseinequality symbol.

–

Solve– 0.6(x – 5) < 15 –

– 0.6x < 12–

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

Add 5 to both side.

Divide each side by 2.

ANSWER

The solutions are all real numbers less than equal to 14. Check by substituting a number less than 14.

Solve the inequality. Graph your solution.

2x – 5 23<–

2x – 5 23.1. < –

2x 28<–

x 14<–

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

Divide the equation by 6.

Simplify.

ANSWER

The solutions are all real numbers less than or equal to 3.5. Check by substituting a number less than 3.5.

Solve the inequality. Graph your solution.

– 6y +5 –16.

2. < –

y 3.5> –

– 6y +5 –16< –

–6–16 –5<–y

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Write original inequality.

Distributive property.

ANSWER

The solutions are all real numbers greater than 20. Check by substituting a number greater than 20.

Solve the inequality.graph your solution

3. –14 ( p –12) > –2

–14 ( p –12) >

12– p + 12 > –8 – p > –8 –

12 p < 20

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.Identify the number of solutions of an inequality

Solve the inequality, if possible. Graph your solution.

5. 5(m + 5) < 5m + 17

SOLUTION

Write original inequality.

Distributive property

25 < 17 Subtract 5m from each side.

ANSWER

There are no solutions because 25 < 17 is false.

5(m + 5) < 5m + 17

5m + 25 < 5m + 17

6.1 & 6.2 Students will be able to graph linear inequalities with one variable.

Identify the number of solutions of an inequality

Write original inequality.

Distributive property

Subtract 8s from each side.

Solve the inequality,if possible. Graph your solution.

1 – 8s –4(2s –1).6. < –

1 – 8s –4(2s –1)< –

1 – 8s 8s + 4< –

1 4< –

ANSWER

All real numbers are solutions because is true. 1 4< –

SOLUTION