15-Second Questions [ 2 points each] 1. What is the sum of the 10 consecutive integers starting from 4 ? 2. Find the least value of \( x \) that satisfies the inequality \( 5-4 x \leq 2 \). 3. The sides of a triangle are 8,12 , and 14 cm long. A new triangle is cor of the sides of the first triangle with segments. What is the perimeter 4. What is the remainder when \( 1-x+2 x^{2}-3 x^{3} \) is divided by \( x+1 \) ? 5. Find the values of \( x \) that satisfy the equation \( |x|+x=0 \).

The sum of 10 consecutive integers starting from 4 can be found using the formula for the sum of an arithmetic series. The integers are 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13. This gives us the sum \( \frac{n}{2} (first + last) = \frac{10}{2} (4 + 13) = 5 \times 17 = 85 \). To solve the inequality \( 5 - 4x \leq 2 \), first, subtract 5 from both sides to get \( -4x \leq -3 \). Dividing both sides by -4 (remember to flip the inequality sign) gives \( x \geq \frac{3}{4} \). So the least value of \( x \) that satisfies the inequality is \( \frac{3}{4} \). The triangle sides of 8 cm, 12 cm, and 14 cm form a valid triangle. The new triangle's sides can be formed by joining the midpoints of the original triangle's sides. The perimeter of the new triangle is half of the perimeter of the original triangle. The original perimeter is \( 8 + 12 + 14 = 34 \), so the perimeter of the new triangle is \( \frac{34}{2} = 17 \) cm. For the polynomial \( 1 - x + 2x^2 - 3x^3 \) divided by \( x + 1 \), we can use the Remainder Theorem. To find the remainder, evaluate the polynomial at \( x = -1 \): \( 1 - (-1) + 2(-1)^2 - 3(-1)^3 = 1 + 1 + 2 + 3 = 7 \). Thus, the remainder is \( 7 \). To solve \( |x| + x = 0 \), recall that the absolute value function can split the equation into two cases. For \( x \geq 0 \), \( |x| = x \), so \( x + x = 0 \) leads to \( 2x = 0 \) and \( x = 0 \). For \( x < 0 \), \( |x| = -x \), making the equation \( -x + x = 0 \), which is always true. Therefore, the solution is \( x = 0 \).