Question

Solve the inequality and graph the solution. -12<2 x-6<4

Short answer

x is any number in the open interval (-3, 5).

Step-by-step solution

Isolate the middle expression

To isolate x, we add 6 to each part of the inequality. The inequality becomes: \(-12 + 6 < 2x - 6 + 6 < 4 + 6\), which simplifies to: \(-6 < 2x < 10\)

Solve for x

To solve for x, we divide each part of the inequality by 2: \(\(-6/2 < 2x/2 < 10/2\)\). This results in: \(-3 < x < 5\). So, x can be any number greater than -3 and less than 5.

Graph the solution

To graph the solution on a number line, we mark -3 and 5 on the number line. We use an open circle to represent -3 and 5 because x cannot equal these values. We shade the area in between -3 and 5 to represent all possible values for x.

Key concepts

Graphing Inequalities

Understanding how to graph inequalities is a fundamental skill in algebra. It allows you to visually represent the range of possible solutions to an inequality. When you have an inequality like (-3 < x < 5), graphing becomes a practical way to showcase all the numbers that satisfy the condition. To graph (-3 < x < 5), you start by drawing a horizontal line, which represents the number line. Since (-3) and (5) are not included in the solutions (indicated by the inequality signs '<'), you will use open circles to mark these numbers. An open circle shows that the endpoint is not part of the solution set.

Next, you shade the region between the open circles to indicate all numbers between (-3) and (5) are the solutions. This visual representation helps students quickly identify valid values for (x) that satisfy the inequality.

Compound Inequalities

Compound inequalities are expressions that combine two or more inequalities. They often look like (-3 < x < 5), expressing that the value of (x) must satisfy two conditions simultaneously. Compound inequalities are essentially shorthand for saying 'x is greater than (-3) AND x is less than (5).' Solving a compound inequality typically involves isolating the variable in the middle, as seen in the given exercise.

To solve the compound inequality, you must perform the same mathematical operation on all three parts. This keeps the inequality balanced. A common mistake when solving compound inequalities is failing to apply changes to all parts of the inequality, leading to an incorrect solution. Remember, you must consider each part of the inequality during each step of your solution process.

Number Line Representation

The number line is a vital tool for representing inequalities because it provides a clear picture of all possible solutions. In our exercise, once we solved the compound inequality (-3 < x < 5), the next step was to represent the solutions on a number line. This graphical representation is straightforward: everything between the two points is shaded, indicating the range of numbers that satisfy the inequality. When dealing with inequalities, it's important to distinguish between 'less than' or 'greater than' (indicated by a '<' or '>' sign) and 'less than or equal to' or 'greater than or equal to' (indicated by a '<=' or '>=' sign).

For 'less than' or 'greater than', use open circles; for 'less than or equal to' or 'greater than or equal to', use closed circles. The closed circle is used when the endpoint is included in the range of solutions. Understanding the difference between these and accurately applying them to the number line ensures precision in interpreting and graphing inequalities.