Solve [tex]|2x+7|=5[/tex] for [tex]x[/tex].
A. [tex]x=-6,1[/tex]
B. [tex]x=-1,6[/tex]
C. [tex]x=-1[/tex]
D. [tex]x=-6,-1[/tex]
E. all real numbers
F. no solution

Split the absolute value equation into two separate equations: 2 x + 7 = 5 and 2 x + 7 = − 5 .
Solve the first equation 2 x + 7 = 5 by subtracting 7 from both sides and then dividing by 2, resulting in x = − 1 .
Solve the second equation 2 x + 7 = − 5 by subtracting 7 from both sides and then dividing by 2, resulting in x = − 6 .
The solutions are x = − 6 and x = − 1 , so the final answer is x = − 6 , − 1 ​ .

Explanation

Understanding the Problem We are given the equation ∣2 x + 7∣ = 5 and we need to find the values of x that satisfy this equation. The absolute value of a number is its distance from zero. Therefore, the expression inside the absolute value, 2 x + 7 , must be either 5 or -5.

Setting up the Equations We set up two equations to solve for x :

2 x + 7 = 5

2 x + 7 = − 5

Solving the First Equation Let's solve the first equation, 2 x + 7 = 5 . We subtract 7 from both sides of the equation:


2 x + 7 − 7 = 5 − 7 2 x = − 2
Now, we divide both sides by 2:
2 2 x ​ = 2 − 2 ​ x = − 1

Solving the Second Equation Next, we solve the second equation, 2 x + 7 = − 5 . We subtract 7 from both sides of the equation:

2 x + 7 − 7 = − 5 − 7 2 x = − 12
Now, we divide both sides by 2:
2 2 x ​ = 2 − 12 ​ x = − 6

Final Answer Therefore, the solutions to the equation ∣2 x + 7∣ = 5 are x = − 1 and x = − 6 .

Examples
Absolute value equations are useful in many real-world scenarios. For example, in manufacturing, if you need to produce parts that are a certain length, say 5 cm, but you allow for a small tolerance of 0.1 cm, then the actual length x of the part must satisfy the equation ∣ x − 5∣ ≤ 0.1 . This means the length can be between 4.9 cm and 5.1 cm. Similarly, in physics, absolute values are used to describe the magnitude of vectors, such as velocity or force, without regard to direction.

Answered by GinnyAnswer | 2025-07-03

The equation ∣2 x + 7∣ = 5 can be solved by breaking it into two separate equations: 2 x + 7 = 5 and 2 x + 7 = − 5 . This results in two solutions: x = − 1 and x = − 6 . Therefore, the answer is A. x = − 6 , − 1 .
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Answered by Anonymous | 2025-07-04