Question

monday - jeff needs at least 75 dollars to buy the bicycle he wants. he has earned $32.75 from mowing lawns and has spent $15 on a bicycle helmet. the inequality ( 75 leq m - 15 + 32.75 ) can be used to find, ( m ), the amount of money he still needs to buy the bicycle. which number line best represents the solution to the inequality? a number line with 26, 27, 28, open circle near 27, arrow left b number line with 56, 57, 58, open circle near 57, arrow right c number line with 26, 27, 28, closed circle near 27, arrow left d number line with 56, 57, 58, closed circle near 57, arrow right tuesday - the solution set for the inequality ( 3\frac{2}{5}x < 21.08 ) is shown on the number line. number line with 5, 6, 7, open circle between 6 and 7, arrow left which value of ( x ) is not in the solution set? a ( -6\frac{3}{5} ) b ( 6\frac{1}{5} ) c ( -\frac{1}{5} ) d ( 6\frac{1}{10} )

Explanation:

Monday Problem (Inequality and Number Line)

Step1: Simplify the inequality

We start with the inequality \( 75 \leq m - 15 + 32.75 \). First, combine the constants on the right side: \( -15 + 32.75 = 17.75 \). So the inequality becomes \( 75 \leq m + 17.75 \).

Step2: Solve for \( m \)

Subtract \( 17.75 \) from both sides of the inequality: \( 75 - 17.75 \leq m \). Calculating \( 75 - 17.75 = 57.25 \). So the solution is \( m \geq 57.25 \).

Step3: Analyze the number lines

• Option A: The number line is around 26 - 28, which is too small. Eliminate A.
• Option B: Open circle at ~57.25? No, our solution is \( m \geq 57.25 \), so we need a closed circle (since it's "greater than or equal to")? Wait, wait, let's check the calculation again. Wait, the original inequality is \( 75 \leq m - 15 + 32.75 \). Wait, maybe I misread the inequality. Wait, the problem says "the inequality \( 75 \leq m - 15 + 32.75 \)". Wait, that would be \( 75 \leq m + 17.75 \). Then \( m \geq 75 - 17.75 = 57.25 \). So we need a number line with a closed circle (since it's "greater than or equal to") at 57.25 and pointing to the right. Let's check the options:
• Option D: Closed circle at ~57.25 (between 57 and 58) and arrow to the right. That matches \( m \geq 57.25 \).

Wait, let's re-express the inequality correctly. Let's re-express the original inequality: \( 75 \leq m - 15 + 32.75 \). Combine like terms: \( m + ( -15 + 32.75 ) = m + 17.75 \). So \( 75 \leq m + 17.75 \). Subtract 17.75: \( m \geq 75 - 17.75 = 57.25 \). So the solution is \( m \geq 57.25 \). So the number line should have a closed dot (since it's "greater than or equal to") at 57.25 and the arrow to the right. Looking at the options:

• Option D: Closed dot between 57 and 58, arrow right. That's correct. So the answer for Monday is D.

Tuesday Problem (Inequality Solution Set)

Step1: Solve the inequality \( 3\frac{2}{5}x < 21.08 \)

First, convert \( 3\frac{2}{5} \) to a decimal: \( 3\frac{2}{5} = 3.4 \). So the inequality is \( 3.4x < 21.08 \).

Step2: Solve for \( x \)

Divide both sides by 3.4: \( x < \frac{21.08}{3.4} \). Calculating \( \frac{21.08}{3.4} = 6.2 \). So the solution set is \( x < 6.2 \).

Step3: Check which value is not in the solution set

We need to find which value is \( \geq 6.2 \).

• Option A: \( -6\frac{3}{5} = -6.6 \), which is less than 6.2. In the solution set.
• Option B: \( 6\frac{1}{5} = 6.2 \), which is equal to 6.2. Since the inequality is \( x < 6.2 \), 6.2 is not in the solution set.
• Option C: \( -\frac{1}{5} = -0.2 \), less than 6.2. In the solution set.
• Option D: \( 6\frac{1}{10} = 6.1 \), less than 6.2. In the solution set.

So the value not in the solution set is B.

Answer:

s:

• Monday: D
• Tuesday: B