Question

Solve the inequality. Then graph the solution set on the real number line. \(1<2 x+3<9\)

Short answer

The solution set for the inequality is \(-1<x<3\). On the number line it can be represented with an open circle at -1 and 3, and a line connecting these two points.

Step-by-step solution

Solving the first inequality

To isolate \(x\) in the first inequality \(1<2x+3\), first subtract 3 from both sides. This gives us \(-2<2x\). Then divide each part by 2. The resulting inequality is \(-1<x\).

Solving the second inequality

To isolate \(x\) in the second inequality \(2x+3<9\), subtract 3 from both sides, which gives \(2x<6\). Dividing each side by 2, we get \(x<3\).

Determining the common solution

The solutions to the two inequalities that were obtained (\(-1<x\) and \(x<3\)) are intersecting, meaning that we are looking for \(x\) values that satisfy both inequalities. The common solution is evident when we consider both inequalities together. The combined inequality which represents the solution to the original problem is \(-1<x<3\).

Graphing the solution

The graph of the solution is a line segment on the number line between -1 and 3 (not including -1 and 3). It can be represented with an open circle at -1 and 3, and a line connecting these two points.

Key concepts

Graphing on the number line

When we talk about graphing on the number line, it is an excellent way to visually represent the solution to an inequality. A number line gives us a linear perspective of the values that satisfy an inequality.

In the problem, we solved the compound inequality, which gave us two components: \(-1 < x < 3\). To graph this solution:

• Draw a horizontal line and mark evenly spaced numbers on it, including -1 and 3.
• Since -1 and 3 are not part of the solution (the inequality is strict), we represent them with open circles on the number line.
• Draw a line connecting these open circles to show all the values between -1 and 3 are valid solutions.

This visual representation lets us see all the possible values of \(x\) that satisfy the inequality. Open circles indicate that the boundary points are not included. These simple steps provide clarity and make it easier to understand the range of values involved.

Compound inequalities

Compound inequalities involve two separate inequalities that are connected by an "and" or an "or". In our case, the problem used "and", which means we want values of \(x\) that satisfy both inequalities at the same time.

Here's how compound inequalities work:

• Each part of the inequality is solved separately. In our example, this means solving \(-1 < x\) and \(x < 3\).
• We then find the intersection of these solutions. In other words, we look for the values of \(x\) that make both inequalities true.

For the example - the solution is the set of all real numbers \(x\) that are greater than -1 and less than 3. Compound inequalities are a way of expressing a range of values that satisfy multiple conditions.

Solving linear inequalities

The process of solving linear inequalities is very similar to solving equations, but with a key difference: when you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

For solving a simple linear inequality:

• Perform the same operations on each side of the inequality as you would with an equation to isolate the variable.
• For the given inequality \(1 < 2x + 3 < 9\), handle each part separately, simplifying step-by-step.
• Subtract or add quantities across the inequality to zero out constants next to the variable.
• Finally, divide or multiply to solve for the variable, keeping in mind to flip the inequality sign if dividing or multiplying by a negative number, although this did not occur in our specific problem.

This method helps to systematically find the range of values that satisfy the inequality, ensuring a step-by-step approach so no errors occur.