Question

example 2
write an inequality that represents each graph.
7.
8.
9.
10.
11.
12.
examples 3–5
solve each inequality.

1. $m - 4 < 3$ 14. $p - 6 \geq 3$
2. $r - 8 \leq 7$ 16. $t - 3 > -8$
3. $b + 2 \geq 4$ 18. $13 > 18 + r$
4. $5 + c \leq 1$ 20. $-23 \geq q - 30$
5. $11 + m \geq 15$ 22. $n - 26 < 4$
6. $8 \leq r - 14$ 24. $-7 > 20 + c$
7. $2a \leq -4 + a$ 26. $z + 4 \geq 2z$

Answer:

Let's solve these inequalities one by one. We'll use the addition or subtraction property of inequalities, which states that adding or subtracting the same number from both sides of an inequality does not change the inequality sign.

Problem 13: \( m - 4 < 3 \)

Step 1: Add 4 to both sides

To isolate \( m \), we add 4 to both sides of the inequality.
\( m - 4 + 4 < 3 + 4 \)

Step 2: Simplify both sides

Simplifying the left side: \( m - 4 + 4 = m \)
Simplifying the right side: \( 3 + 4 = 7 \)
So, \( m < 7 \)

Problem 14: \( p - 6 \geq 3 \)

Step 1: Add 6 to both sides

To isolate \( p \), we add 6 to both sides of the inequality.
\( p - 6 + 6 \geq 3 + 6 \)

Step 2: Simplify both sides

Simplifying the left side: \( p - 6 + 6 = p \)
Simplifying the right side: \( 3 + 6 = 9 \)
So, \( p \geq 9 \)

Problem 15: \( r - 8 \leq 7 \)

Step 1: Add 8 to both sides

To isolate \( r \), we add 8 to both sides of the inequality.
\( r - 8 + 8 \leq 7 + 8 \)

Step 2: Simplify both sides

Simplifying the left side: \( r - 8 + 8 = r \)
Simplifying the right side: \( 7 + 8 = 15 \)
So, \( r \leq 15 \)

Problem 16: \( t - 3 > -8 \)

Step 1: Add 3 to both sides

To isolate \( t \), we add 3 to both sides of the inequality.
\( t - 3 + 3 > -8 + 3 \)

Step 2: Simplify both sides

Simplifying the left side: \( t - 3 + 3 = t \)
Simplifying the right side: \( -8 + 3 = -5 \)
So, \( t > -5 \)

Problem 17: \( b + 2 \geq 4 \)

Step 1: Subtract 2 from both sides

To isolate \( b \), we subtract 2 from both sides of the inequality.
\( b + 2 - 2 \geq 4 - 2 \)

Step 2: Simplify both sides

Simplifying the left side: \( b + 2 - 2 = b \)
Simplifying the right side: \( 4 - 2 = 2 \)
So, \( b \geq 2 \)

Problem 18: \( 13 > 18 + r \)

Step 1: Subtract 18 from both sides

To isolate \( r \), we subtract 18 from both sides of the inequality.
\( 13 - 18 > 18 + r - 18 \)

Step 2: Simplify both sides

Simplifying the left side: \( 13 - 18 = -5 \)
Simplifying the right side: \( 18 + r - 18 = r \)
So, \( -5 > r \) or \( r < -5 \)

Problem 19: \( 5 + c \leq 1 \)

Step 1: Subtract 5 from both sides

To isolate \( c \), we subtract 5 from both sides of the inequality.
\( 5 + c - 5 \leq 1 - 5 \)

Step 2: Simplify both sides

Simplifying the left side: \( 5 + c - 5 = c \)
Simplifying the right side: \( 1 - 5 = -4 \)
So, \( c \leq -4 \)

Problem 20: \( -23 \geq q - 30 \)

Step 1: Add 30 to both sides

To isolate \( q \), we add 30 to both sides of the inequality.
\( -23 + 30 \geq q - 30 + 30 \)

Step 2: Simplify both sides

Simplifying the left side: \( -23 + 30 = 7 \)
Simplifying the right side: \( q - 30 + 30 = q \)
So, \( 7 \geq q \) or \( q \leq 7 \)

Problem 21: \( 11 + m \geq 15 \)

Step 1: Subtract 11 from both sides

To isolate \( m \), we subtract 11 from both sides of the inequality.
\( 11 + m - 11 \geq 15 - 11 \)

Step 2: Simplify both sides

Simplifying the left side: \( 11 + m - 11 = m \)
Simplifying the right side: \( 15 - 11 = 4 \)
So, \( m \geq 4 \)

Problem 22: \( n - 26 < 4 \)

Step 1: Add 26 to both sides

To isolate \( n \), we add 26 to both sides of the inequality.
\( n - 26 + 26 < 4 + 26 \)

Step 2: Simplify both sides

Simplifying the left side: \( n - 26 + 26 = n \)
Simplifying the right side: \( 4 + 26 = 30 \)
So, \( n < 30 \)

Problem 23: \( 8 \leq r - 14 \)

Step 1: Add 14 to both sides

To isolate \( r \), we add 14 to both sides of the inequality.
\( 8 + 14 \leq r - 14 + 14 \)

Step 2: Simplify both sides

Simplifying the left side: \( 8 + 14 = 22 \)
Simplifying the right side: \( r - 14 + 14 = r \)
So, \( 22 \leq r \) or \( r \geq 22 \)

Problem 24: \( -7 > 20 + c \)

Step 1: Subtract 20 from both sides

To isolate \( c \), we subtract 20 from both sides of the inequality.
\( -7 - 20 > 20 + c - 20 \)

Step 2: Simplify both sides

Simplifying the left side: \( -7 - 20 = -27 \)
Simplifying the right side: \( 20 + c - 20 = c \)
So, \( -27 > c \) or \( c < -27 \)

Problem 25: \( 2a \leq -4 + a \)

Step 1: Subtract \( a \) from both sides

To isolate \( a \), we subtract \( a \) from both sides of the inequality.
\( 2a - a \leq -4 + a - a \)

Step 2: Simplify both sides

Simplifying the left side: \( 2a - a = a \)
Simplifying the right side: \( -4 + a - a = -4 \)
So, \( a \leq -4 \)

Problem 26: \( z + 4 \geq 2z \)

Step 1: Subtract \( z \) from both sides

To isolate \( z \), we subtract \( z \) from both sides of the inequality.
\( z + 4 - z \geq 2z - z \)

Step 2: Simplify both sides

Simplifying the left side: \( z + 4 - z = 4 \)
Simplifying the right side: \( 2z - z = z \)
So, \( 4 \geq z \) or \( z \leq 4 \)

Final Answers:

1. \( \boldsymbol{m < 7} \)
2. \( \boldsymbol{p \geq 9} \)
3. \( \boldsymbol{r \leq 15} \)
4. \( \boldsymbol{t > -5} \)
5. \( \boldsymbol{b \geq 2} \)
6. \( \boldsymbol{r < -5} \)
7. \( \boldsymbol{c \leq -4} \)
8. \( \boldsymbol{q \leq 7} \)
9. \( \boldsymbol{m \geq 4} \)
10. \( \boldsymbol{n < 30} \)
11. \( \boldsymbol{r \geq 22} \)
12. \( \boldsymbol{c < -27} \)
13. \( \boldsymbol{a \leq -4} \)
14. \( \boldsymbol{z \leq 4} \)