Question

Solve the inequality. Write a sentence that describes the solution. $$ -4<2+x<1 $$

Short answer

The solution to the inequality -4 < 2 + x < 1 is -6 < x < -1.

Step-by-step solution

Rearrange the inequality

The inequality can be broken down to two simple inequalities: \n1. -4 < 2 + x, and \n2. 2 + x < 1. For both inequalities, it is necessary to subtract 2 in order to isolate x.

Solve for x for the first inequality

For the first inequality, -4 < 2 + x, we can subtract 2 from both sides to get: -4 - 2 < 2 + x - 2. Simplifying, this leads to -6 < x or x > -6.

Solve for x for the second inequality

We can do the same for the second inequality, 2 + x < 1. Subtract 2 from both sides again to get: 2 + x - 2 < 1 - 2. Simplifying this results in x < -1.

Determine the solution range for x

Combining the two inequalities from Steps 2 and 3, the solution range for x that satisfy the inequality is -6 < x < -1

Key concepts

Compound Inequalities

Understanding compound inequalities involves dealing with two related inequalities at once, linked by 'and' or 'or'. When the inequalities are connected by 'and', it means the solution must satisfy both inequalities simultaneously. With the inequality \[\[\begin{align*}-4<2+x<1\text{,}\text{ two separate but connected inequalities emerge: }-4<2+x \text{ and }2+x<1.\end{align*}\]\]To decipher the solution range, both conditions must be met. It’s like finding a common ground between the two, an overlap where all parts agree. The process of solving compound inequalities is systematic: separate the inequalities, isolate the variable in each, and identify the overlap in their solutions.

Isolate the Variable

To isolate the variable in inequalities, similar steps to solving equations are followed, but with special considerations. The goal is to get the variable, in our example 'x', all by itself on one side of the inequality sign. For the inequality\[\[\begin{align*}-4<2+x<1,\text{ isolation requires subtraction of 2 from all parts. This yields two separate inequalities: }-6

Inequality Solution Range

The solution range of an inequality is the set of all values that satisfy the inequality's conditions. After isolating the variables, in our case with the inequality \[\[\begin{align*}-6

Inequality Solution Range

The solution range of an inequality is the set of all values that satisfy the inequality's conditions. After isolating the variables, in our case with the inequality \[\[\begin{align*}-6