2-5 Absolute Value Equations

Michael is trying to solve a problem in his math homework: |2x - 3| = 5. He needs to find the value of 'x'. Ava and Elijah, his classmates, gave him different solutions. If Abigail, Anika, and Samuel were to help him choose the correct solution, which one should they suggest?

• Ava's solution: x = 4, -1

• Elijah's solution: x = 1, -2

• Both are correct

• Neither are correct

Answer explanation

The correct solution to the question is Ava's solution: x = 4, -1. This is because when you solve the equation |2x - 3| = 5, you get two possible solutions for x, which are 4 and -1. Therefore, Ava's solution is the correct one.

William, Aiden, Mason, Avery, Benjamin, and Maya are in a math competition. They are given a real-life problem to solve: A company's profit (P) is given by the equation |4 - 3x| = 7, where x is the number of products sold. Can you help them find the possible values of x?

• x = 2, -5/3

• x = -1, 11/3

• x = 5, -7/3

• x = 0, 7/3

Answer explanation

The equation |4 - 3x| = 7 can be solved as 4 - 3x = 7 or 4 - 3x = -7. Solving these gives x = -1 and x = 11/3 respectively. Hence, the correct answer is x = -1, 11/3.

Harper, Ava, and Luna are trying to divide a basket of apples equally among themselves. They came up with the equation: |5x + 2| = 3 to represent the number of apples each of them should get. Can you help them solve it?

• x = 3/5, -1

• x = 1/5, 2

• x = -1/5, 1

• x = 1/5, -1

Answer explanation

The question asks to solve the equation |5x + 2| = 3. The absolute value equation can be solved as 5x + 2 = 3 and 5x + 2 = -3. Solving these gives x = 1/5 and x = -1 respectively. Hence, the correct answer is x = 1/5, -1.

Aiden has a 2-meter long rope for a school project. He needs to cut it into pieces, each measuring 4x meters long. If the absolute difference between the original length of the rope and the total length of the cut pieces is 10 meters, what could be the value of x?

• x = 2, -3

• x = 4, -5

• x = 1, -4

• x = -2, 3

Answer explanation

The question asks for the value of x when a 2-meter rope is cut into pieces of 4x meters each, and the absolute difference between the original and total length of the pieces is 10 meters. The correct answer is x = -2, 3. This means that the pieces of rope are either 3 meters or -2 meters long, which when subtracted from the original length, gives a difference of 10 meters.

Isla, Abigail, and Emma are trying to find the number of apples (x) they need to buy to balance the weight of their basket. The absolute difference between the weight of 3x apples and 1 apple is 2 pounds. Solve the absolute value equation: |3x - 1| = 2

• x > 1 or x < 1

• x > 1/2 or x < 1/2

• x > 2/3 or x < 2/3

• x > 1/3 or x < 1/3

Answer explanation

To solve the absolute value equation |3x - 1| = 2, we consider two cases: 3x - 1 = 2 and 3x - 1 = -2. Solving these equations, we get x = 1/3 and x = -1/3, respectively. Therefore, the solution is x > 1/3 or x < 1/3, which matches the correct choice.

Michael has 6 apples and wants to give 2x apples to Kai and Scarlett. Sophia, Elijah, and William are watching. If the absolute value of the difference between the number of apples Michael has left and the number of apples he gives away is 8, find the value of x.

• x = -1, 7

• x = 2, 7

• x = 1, 7

• x = -1, -7

Answer explanation

In the question, Michael has 6 apples and wants to give 2x apples to Kai and Scarlett. The absolute value of the difference between the remaining apples and the given apples is 8. By solving the equation |6 - 2x| = 8, we get two possible values for x: -1 and 7. Therefore, the correct choice is x = -1, 7.

Charlotte and James are standing on a number line. The distance between them is represented by the absolute value equation: |4x + 5| = 9. If x represents the position of Scarlett on the number line, what are the possible values of x for Scarlett to be equidistant from both Charlotte and James?

• x = 2, -4.5

• x = -1, 3.5

• x = 1, -3.5

• x = 0, -2.5

Answer explanation

The absolute value equation |4x + 5| = 9 can be solved as 4x + 5 = 9 or 4x + 5 = -9. Solving these gives x = 1 and x = -3.5 respectively. These are the positions on the number line where Scarlett is equidistant from both Charlotte and James.

Aria has 7 apples and wants to give away 2x apples to her friends Anika and Mia. If the absolute value of the remaining apples is 1, solve for x:

|7 - 2x| = 1

• x < 2.5 or x > 2.5

• x < 1.5 or x > 1.5

• x < 4.5 or x > 4.5

• x < 3.5 or x > 3.5

Answer explanation

The question asks to solve for x in the equation |7 - 2x| = 1. This absolute value equation has two solutions: 7 - 2x = 1 and 7 - 2x = -1. Solving these gives x = 3 and x = 4 respectively. Therefore, the solution is x < 3.5 or x > 3.5.

Liam has a piece of rope that is 3 times longer than the height of a tree in Abigail's garden, minus 4 inches. If the absolute length of the rope is 6 inches, what could be the height of the tree?

• The tree could be 2/3 inches or -10/3 inches tall

• The tree could be 4/3 inches or -8/3 inches tall

• The tree could be 10/3 inches or -2/3 inches tall

• The tree could be 5/3 inches or -7/3 inches tall

Answer explanation

The question states that Liam's rope is 3 times the height of the tree minus 4 inches and its length is 6 inches. By setting up the equation 3x - 4 = 6, we can solve for x, which represents the height of the tree. The solutions are 10/3 inches and -2/3 inches, which matches the correct answer.

Emma is at a grocery store and she sees a discount offer. The offer is such that the discount is the absolute value of the equation: |2 - 5x| = 4. If Emma wants to maximize her discount, can you help her find the value of x?

• x = -2/5, 6/5

• x = -4/5, 2/5

• x = -2/5, 6/5

• x = -1/5, 7/5

Answer explanation

The question asks for the value of x that maximizes the discount, which is given by the equation |2 - 5x| = 4. Solving this equation gives two solutions: x = -2/5 and x = 6/5. Therefore, the correct answer is 'x = -2/5, 6/5'.