1. Sam solved the following problem incorrectly: -4x + 5 > 29 -4x + 5 - 5 > 29 - 5 -4x > -24 -4x/-4 > -24/-4 x > 6 What were his two mistakes? (3 points) 2. Hillary buys 3 pounds of Gala apples and some Granny Smith apples. Both kinds of apples cost $4.50 per pound. The total cost is $24.75. How many pounds of Granny Smith apples did Hillary buy? Write and solve an equation with a variable. Show your work. (4 points) 3. Determine whether each equation has one solution, infinitely many solutions, or no solution. (3 points) - 6x + 8 = 6(x + 2) - 10x = 15 + 5x - x + 11 = 8x + 11 - 7x

1. Sam made two mistakes when solving the inequality:

First mistake: When he divided by -4 on both sides of the inequality, he did not reverse the inequality symbol. When you divide or multiply both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed. The correct operation should be: \(-4x < 24\) when divided by -4 because \(x\) should be less than \(-6\), not greater than 6.

Second mistake: Even though his division mistake led to a wrong inequality, he also miscalculated the division of -24 by -4. The result of \(-24/-4\) is 6, not -6 as he might imply with his incorrect inequality direction. So, if he did not make the first mistake, he should have found that \(x < 6\).

2. To find out how many pounds of Granny Smith apples Hillary bought, let’s set up an equation.

Let \( x \) represent the number of pounds of Granny Smith apples.

The cost for Gala apples is \( 3 \text{ pounds} \times $4.50/\text{pound} = $13.50 \). The cost for Granny Smith apples is \( x \text{ pounds} \times $4.50/\text{pound} = $4.50x \).

The total cost is $24.75, so the equation to find \( x \) is: \( $13.50 + $4.50x = $24.75 \)

Now, solve for \( x \): \( $4.50x = $24.75 - $13.50 \) \( $4.50x = $11.25 \)

Divide both sides by $4.50 to find the number of pounds: \( x = $11.25 / $4.50 \) \( x = 2.5 \)

Hillary bought 2.5 pounds of Granny Smith apples.

3. For each equation:

- \( -6x + 8 = 6(x + 2) \)

Distribute the right side and then simplify: \( -6x + 8 = 6x + 12 \)

Subtract \( 6x \) from both sides and subtract \( 8 \) from both sides: \( -6x - 6x + 8 - 8 = 12 - 8 \)

\( 0 ≠ 4 \)

This equation has no solution since the variables cancel out and we are left with a false statement.

- \( 10x = 15 + 5x \)

Subtract \( 5x \) from both sides of the equation: \( 10x - 5x = 15 \)

\( 5x = 15 \)

Divide both sides by \( 5 \): \( x = 3 \)

This equation has one solution, \( x = 3 \).

- \( x + 11 = 8x + 11 - 7x \)

Combine like terms on the right side: \( x + 11 = x + 11 \)

Subtract \( x \) from both sides: \( x - x + 11 = x - x + 11 \)

\( 11 = 11 \)

This equation has infinitely many solutions since the variables cancel out and we are left with a true statement. Any value of \( x \) will satisfy this equation.

Extra: Understanding the concepts:

Inequalities are similar to equations but instead of an equal sign, they have inequality symbols like \(>\), \(<\), \(≥\), \(≤\). When you perform operations like multiplying or dividing by a negative number in inequalities, it's crucial to remember that the direction of the inequality changes. This is because the negative sign reverses the order of numbers on the number line. Therefore, inequalities require careful manipulation to maintain their true meaning.

When solving word problems like the apple cost problem in question 2, a common strategy is to set up a variable to represent the unknown quantity, which in this case is the number of pounds of Granny Smith apples. By creating an equation based on the given prices and total cost, you can then solve for the variable using basic algebraic techniques.

The third question deals with understanding when an equation might have one, no, or infinitely many solutions. The process usually involves manipulating the equation to try and isolate the variable. If you end up with a true statement without the variable (like \(11=11\)), the equation has infinitely many solutions. If you get a false statement (like \(0 ≠ 4\)), then the equation has no solutions. Finally, if you can solve for the variable and find a specific number (like \(x=3\)), the equation has one solution.