3-5 Solving Inequalities with Variables on Both Sides Warm Up Solve each equation. 1. 2x = 7x + 15...

3-5 Solving Inequalities with Variables on Both Sides

Warm UpSolve each equation. 1. 2x = 7x + 15

2.

5. Solve and graph 5(2 – b) > 52.

3. 2(3z + 1) = –2(z + 3)

4. 3(p – 1) = 3p + 2

x = –3

b < –3

–5 –3 –2 –1–4 0–6

3y – 21 = 4 – 2y y = 5

z = –1

no solution


Additional Example 1A: Solving Inequalities with Variables on Both Sides

Solve the inequality and graph the solutions.

y ≤ 4y + 18

y ≤ 4y + 18–y –y

0 ≤ 3y + 18

–18 – 18

–18 ≤ 3y

To collect the variable terms on one side, subtract y from both sides.

Since 18 is added to 3y, subtract 18 from both sides to undo the addition.

Since y is multiplied by 3, divide both sides by 3 to undo the multiplication.


Additional Example 1A: Continued


y ≤ 4y + 18

–6 ≤ y (or y –6)

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {y:y ≥ –6}.


4m – 3 < 2m + 6To collect the variable terms on one

side, subtract 2m from both sides.–2m – 2m

2m – 3 < + 6

Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction.

+ 3 + 3

2m < 9

Since m is multiplied by 2, divide both sides by 2 to undo the multiplication.

Additional Example 1B: Solving Inequalities with Variables on Both Sides



4m – 3 < 2m + 6

Additional Example 1B Continued


4 5 6

The solution set is {m:m }.


Solve the inequality and graph the solutions. Check your answer.

Check It Out! Example 1a

4x ≥ 7x + 6

4x ≥ 7x + 6–7x –7x

–3x ≥ 6

x ≤ –2

To collect the variable terms on one side, subtract 7x from both sides.

Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {x:x ≤ –2}.


Check It Out! Example 1a Continued


4x ≥ 7x + 6

Check

Check the endpoint, –2.

–8 –14 + 6

4(–2) 7(–2) + 6

–8 –8

4x = 7x + 6

Check a number less than –2.

–12 ≥ –15

4x ≥ 7x + 6

4(–3) ≥ 7(–3) + 6–12 ≥ –21 + 6



Check It Out! Example 1b

5t + 1 < –2t – 6

5t + 1 < –2t – 6+2t +2t

7t + 1 < –6– 1 < –1

7t < –77t < –77 7

t < –1

–5 –4 –3 –2 –1 0 1 2 3 4 5

To collect the variable terms on one side, add 2t to both sides.

Since 1 is added to 7t, subtract 1 from both sides to undo the addition.

Since t is multiplied by 7, divide both sides by 7 to undo the multiplication.

The solution set is {t:t < –1}.


Additional Example 3A: Simplify Each Side Before Solving


2(k – 3) > 6 + 3k – 3

2(k – 3) > 3 + 3k Distribute 2 on the left side of the inequality.

2k + 2(–3) > 3 + 3k

2k – 6 > 3 + 3k–2k – 2k

–6 > 3 + k

To collect the variable terms, subtract 2k from both sides.

–3 –3

–9 > k

Since 3 is added to k, subtract 3 from both sides to undo the addition.


Additional Example 3A Continued

–9 > k

–12 –9 –6 –3 0 3


The solution set is {k:k < –9}.

2(k – 3) > 6 + 3k – 3


Check It Out! Example 3a


5(2 – r) ≥ 3(r – 2)

5(2 – r) ≥ 3(r – 2)

5(2) – 5(r) ≥ 3(r) + 3(–2)

10 – 5r ≥ 3r – 6+6 +6

16 − 5r ≥ 3r+ 5r +5r

16 ≥ 8r

Distribute 5 on the left side of the inequality and distribute 3 on the right side of the inequality.

Since 6 is subtracted from 3r, add 6 to both sides to undo the subtraction.

Since 5r is subtracted from 16 add 5r to both sides to undo the subtraction.


Check It Out! Example 3a Continued

–6 –2 0 2–4 4

16 ≥ 8r Since r is multiplied by 8, divide both sides by 8 to undo the multiplication.

2 ≥ r


The solution set is {r:r ≤ 2}.


Lesson Quiz: Part I

Solve each inequality and graph the solutions.

1. t < 5t + 24 t > –6

2. 5x – 9 ≤ 4.1x – 81 x ≤ –80

b < 133. 4b + 4(1 – b) > b – 9


Lesson Quiz: Part II

4. Rick bought a photo printer and supplies for $186.90, which will allow him to print photos for $0.29 each. A photo store charges $0.55 to print each photo. How many photos must Rick print before his total cost is less than getting prints made at the photo store?

Rick must print more than 718 photos.


Lesson Quiz: Part III

Solve each inequality.

5. 2y – 2 ≥ 2(y + 7)

6. 2(–6r – 5) < –3(4r + 2)

all real numbers

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3-5 Solving Inequalities with Variables on Both Sides Warm Up Solve each equation. 1. 2x = 7x + 15...

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