Problem 58 - College Algebra

Question

Solve compound inequality. \(-6 \leq \frac{1}{2} x-4<-3\)

Step-by-Step Solution

Answer

The solution of the compound inequality is \(-4 \leq x<2\).

Step 1: Solve the First Inequality

To solve the inequality \(-6 \leq \frac{1}{2}x-4\), start by adding 4 to both sides to isolate the term that includes \(x\): \(-6+4 \leq \frac{1}{2}x\). So, \(-2 \leq \frac{1}{2}x\). Next, multiply both sides by 2 to solve for \(x\). This gives \(-2*2 \leq x\), resulting in \(-4 \leq x\).

Step 2: Solve the Second Inequality

To solve the inequality \(\frac{1}{2} x-4<-3\), add 4 to both sides: \(\frac{1}{2}x<-3+4\), which simplifies to \(\frac{1}{2}x<1\). Multiplying both sides by 2, we obtain: \(x<1*2\) or \(x<2\).

Step 3: Combine the Results

The solution to the compound inequality is the intersection of the solutions to the individual inequalities. This is given by the highest lower bound and the lowest upper bound of the two inequalities. That renders the solution \(-4 \leq x<2\). This means \(x\) is greater than or equal to -4 and less than 2.

Key Concepts

Isolation of Variables

When solving inequalities, the goal is often to isolate the variable, typically represented by \(x\). Isolating the variable means performing operations that "unwrap" the variable from the other numbers and symbols surrounding it. To do this in a step-by-step process, follow these guidelines:

• Identify the term that contains the variable you want to solve. In our case, we start with \(\frac{1}{2}x\), which is surrounded by numbers.


• Use arithmetic operations, such as addition, subtraction, multiplication, or division, to move numbers away from the variable. For instance, in the inequality \(-6 \leq \frac{1}{2}x - 4\), you first add 4 to both sides to begin isolating \(x\).


• Continue simplifying until the variable is by itself on one side of the inequality symbol. In the step with \(\frac{1}{2}x\), multiplying both sides by 2 helps to isolate \(x\).

Through careful isolation of variables, you turn complex inequalities into simpler, solvable forms, allowing you to clearly see the possible range of values for \(x\).

Inequality Solving

Inequalities are expressions that involve comparison using symbols like \(\leq, <, >, \) and \( \geq \). Solving inequalities is similar to solving equations, with a few important differences that you need to keep in mind:

• When you add or subtract a number from both sides, the inequality direction remains unchanged. For example, adding 4 to both sides of \(-6 \leq \frac{1}{2}x-4\) maintains the same direction.


• Multiplying or dividing both sides by a positive number also keeps the inequality direction unchanged; multiplying by 2 doesn't flip the sign in \(-2 \leq \frac{1}{2}x\) becoming \(-4 \leq x\).


• However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign. This rule is vital to remember!

Through understanding these rules, solving inequalities becomes an easier task, and you avoid common pitfalls that occur with sign direction changes.

Intersection of Solutions

The intersection of solutions is an essential concept when working with compound inequalities. Compound inequalities involve two separate inequalities being solved simultaneously, and the solution to the compound inequality is where both individual solutions overlap.
To determine the intersection:

• First, solve each part of the compound inequality separately, just like any other inequality. For instance, solve \(-6 \leq \frac{1}{2}x-4\) and \(\frac{1}{2}x-4<-3\).


• Look for common solutions. Ask yourself: "Where does the solution to one inequality overlap with the solution to the other?"


• The intersection is established by determining the highest lower limit and the lowest upper limit of \(x\). In this example, the outcomes \(-4 \leq x\) and \(x < 2\) intersect at \(-4 \leq x < 2\).

By understanding the concept of intersection, you gain a fuller picture of the range of possible values that satisfy the entire compound inequality. Mastering intersections ensures comprehensive solutions.