Question

Solve the inequality. Write a sentence that describes the solution. $$ -4 \leq-3 x-13 \leq 26 $$

Short answer

The solution to the inequality is \( -3 \geq x \geq -13 \).

Step-by-step solution

Prediction

The inequality is a compound inequality which consists of two inequalities. The goal is to solve for 'x'. Begin by handling each inequality separately.

Solve the Left Inequality

Consider the left inequality: \(-4 \leq -3x -13\). In order to isolate 'x', begin by adding 13 to both sides of the inequality to get: \(9 \leq -3x\). Now divide both sides of the inequality by -3. Notice that since -3 is a negative number, when dividing by a negative, the direction of the inequality changes. Therefore, the result inequality would be: \(-3 \geq x\) or \(x \leq -3\).

Solve the Right Inequality

Next, consider the right inequality: \(-3x - 13 \leq 26\). Similar to step 2, begin by adding 13 to both sides of the inequality to get: \(-3x \leq 39\). Dividing both sides by -3 will give \(\frac{-39}{3} \geq x\), which simplifies to: \(x \geq -13\).

Combine the Solutions

Both the solved inequalities from step 2 and step 3 have to be true for the original compound inequality to be true. Therefore, if you combine both inequalities you get: \(-3 >= x >= -13\).

Key concepts

Compound Inequality

A compound inequality combines two or more inequalities into one statement. It can be viewed as a conjunction (both must be true) or a disjunction (one or the other must be true). In our exercise, we're dealing with a conjunction since we have two inequalities linked by an 'and' condition: \(-4 \leq -3x - 13 \leq 26\).

When solving a compound inequality, each part of the inequality must be solved independently while keeping in mind that the final solution must satisfy all individual inequalities. The solutions are then combined to find the set of values that satisfy the entire compound inequality.

Isolate Variable

To solve for a variable means to isolate it on one side of the inequality or equation. This is done through a series of algebraic manipulations, where we perform the same operations on both sides of the inequality to maintain balance. For inequalities, always remember that unlike equations, multiplying or dividing by a negative number reverses the direction of the inequality.

In the given exercise, isolating 'x' involves two main steps for each part of the compound inequality: adding 13 to both sides to eliminate the constant term, followed by dividing by -3 to solve for 'x'. But crucially, when you divide by -3, you must reverse the inequality to maintain the correct relationship between the numbers.

Direction of Inequality

The direction of the inequality tells us how values on one side compare to those on the other side. The symbols \(<\), \(>\), \(\leq\), and \(\geq\) indicate less than, greater than, less than or equal to, and greater than or equal to, respectively.

When multiplying or dividing both sides of an inequality by a negative number, the inequality flips. This is crucial, as forgetting to reverse the inequality will lead to incorrect solutions. In the exercise we are analyzing, dividing the inequalities by -3 necessitates inverting their direction. This is to preserve the logical condition that's being expressed, since multipliers have the inherent property of reversing relational expressions when negative.

Algebraic Manipulation

Algebraic manipulation involves performing operations to both sides of an equation or inequality to rearrange its terms and solve for a variable. The most important principle to remember is that whatever you do to one side, you must do to the other to maintain equality (or in the case of inequalities, the correct order).

Algebraic manipulations often involve adding, subtracting, multiplying, or dividing both sides by the same value. It also includes simplifying expressions by combining like terms or factoring. In the given compound inequality, we use algebraic manipulation to isolate 'x' and simplify the inequality, culminating in the conclusion that if both inequalities are satisfied, the solution set for 'x' falls within a specific range.