define a variable, write an inequality, and solve each problem. check your solution.
61. seven more than a number is less than or equal to -18.
62. twenty less than a number is at least 15.
63. a number plus 2 is at most 1.
64. one eighth of a number is less than or equal to 3.
65. negative twelve times a number is no more than 84.
66. eight times a number is at least 16.

Question

Accepted Answer

Let's solve each problem one by one. We'll start with problem 61.

### Problem 61: Seven more than a number is less than or equal to -18.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"Seven more than a number" means $ x + 7 $. "Is less than or equal to -18" gives the inequality $ x + 7 \leq -18 $.
## Step 3: Solve the inequality
Subtract 7 from both sides: $ x + 7 - 7 \leq -18 - 7 $, so $ x \leq -25 $.
## Step 4: Check the solution
Let's pick a number less than or equal to -25, say $ x = -26 $. Then $ -26 + 7 = -19 $, and $ -19 \leq -18 $ is true. Another check: $ x = -25 $, $ -25 + 7 = -18 $, and $ -18 \leq -18 $ is also true.

### Problem 62: Twenty less than a number is at least 15.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"Twenty less than a number" is $ x - 20 $. "Is at least 15" means $ x - 20 \geq 15 $.
## Step 3: Solve the inequality
Add 20 to both sides: $ x - 20 + 20 \geq 15 + 20 $, so $ x \geq 35 $.
## Step 4: Check the solution
Pick $ x = 35 $: $ 35 - 20 = 15 $, and $ 15 \geq 15 $ is true. Pick $ x = 36 $: $ 36 - 20 = 16 $, and $ 16 \geq 15 $ is true.

### Problem 63: A number plus 2 is at most 1.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"A number plus 2" is $ x + 2 $. "Is at most 1" gives $ x + 2 \leq 1 $.
## Step 3: Solve the inequality
Subtract 2 from both sides: $ x + 2 - 2 \leq 1 - 2 $, so $ x \leq -1 $.
## Step 4: Check the solution
Pick $ x = -1 $: $ -1 + 2 = 1 $, and $ 1 \leq 1 $ is true. Pick $ x = -2 $: $ -2 + 2 = 0 $, and $ 0 \leq 1 $ is true.

### Problem 64: One eighth of a number is less than or equal to 3.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"One eighth of a number" is $ \frac{1}{8}x $. "Is less than or equal to 3" gives $ \frac{1}{8}x \leq 3 $.
## Step 3: Solve the inequality
Multiply both sides by 8: $ 8 \times \frac{1}{8}x \leq 3 \times 8 $, so $ x \leq 24 $.
## Step 4: Check the solution
Pick $ x = 24 $: $ \frac{1}{8} \times 24 = 3 $, and $ 3 \leq 3 $ is true. Pick $ x = 20 $: $ \frac{1}{8} \times 20 = 2.5 $, and $ 2.5 \leq 3 $ is true.

### Problem 65: Negative twelve times a number is no more than 84.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"Negative twelve times a number" is $ -12x $. "Is no more than 84" means $ -12x \leq 84 $.
## Step 3: Solve the inequality
Divide both sides by -12. Remember, when dividing by a negative number, the inequality sign flips. So $ \frac{-12x}{-12} \geq \frac{84}{-12} $, which simplifies to $ x \geq -7 $.
## Step 4: Check the solution
Pick $ x = -7 $: $ -12 \times (-7) = 84 $, and $ 84 \leq 84 $ is true. Pick $ x = -6 $: $ -12 \times (-6) = 72 $, and $ 72 \leq 84 $ is true.

### Problem 66: Eight times a number is at least 16.
# Explanation:
## Step 1: Define the variable
Let $ x $ be the number.
## Step 2: Write the inequality
"Eight times a number" is $ 8x $. "Is at least 16" gives $ 8x \geq 16 $.
## Step 3: Solve the inequality
Divide both sides by 8: $ \frac{8x}{8} \geq \frac{16}{8} $, so $ x \geq 2 $.
## Step 4: Check the solution
Pick $ x = 2 $: $ 8 \times 2 = 16 $, and $ 16 \geq 16 $ is true. Pick $ x = 3 $: $ 8 \times 3 = 24 $, and $ 24 \geq 16 $ is true.

# Answers:
61. $ x \leq -25 $
62. $ x \geq 35 $
63. $ x \leq -1 $
64. $ x \leq 24 $
65. $ x \geq -7 $
66. $ x \geq 2 $

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QUESTION IMAGE

Question

Explanation:

Problem 61: Seven more than a number is less than or equal to -18.

Step 1: Define the variable

Step 2: Write the inequality

Step 3: Solve the inequality

Step 4: Check the solution

Problem 62: Twenty less than a number is at least 15.

Step 1: Define the variable

Step 2: Write the inequality

Step 3: Solve the inequality

Step 4: Check the solution

Problem 63: A number plus 2 is at most 1.

Step 1: Define the variable

Step 2: Write the inequality

Step 3: Solve the inequality

Step 4: Check the solution

Problem 64: One eighth of a number is less than or equal to 3.

Answer: