Question

Solve the inequality and graph its solution. \(-4+x<20\)

Short answer

The solution for the inequality \(-4+x<20\) is \(x<24\). On a graph, this includes all numbers less than 24, but not 24 itself.

Step-by-step solution

Isolate the Variable

The first step here is to isolate x. To do this, we'll need to get rid of the -4 on the left side of the inequality. We do this by adding 4 to both sides, remember whatever you do on one side must be done on the other side also. This gives us \(x<20+4\).

Solve the Inequality

Next step is to simplify the inequality. It involves adding the numbers on the right side of the inequality to get \(x<24\).

Graph the Solution

The final step is to graph the solution. On a number line, you would mark a open circle at 24 (because x is less than 24, not less than or equal to) and shade all the numbers to the left of 24.

Key concepts

Isolate the Variable

Isolating the variable is an essential first step in solving any algebraic inequality. It means to rearrange the inequality so that the variable we are solving for is by itself on one side of the inequality symbol.

To achieve this, we use inverse operations to eliminate other numbers or variables from the same side as the variable. In the example provided, (-4+x<20), we isolate x by adding 4 to both sides of the inequality, resulting in an equivalent inequality, (x<24), where x stands alone.

Remember, it is important that whatever operation is applied to one side of the inequality, the same must be done to the other side to maintain the balance of the equation. Moreover, there is a special rule in inequalities: reversing the inequality symbol when multiplying or dividing by a negative number is crucial to keep the inequality valid.

Graphing Inequalities

Graphing inequalities on a number line provides a visual representation of the set of all possible solutions. For the given inequality, (x<24), the graph would consist of a number line where we place an open circle at the number 24 to indicate that this number is not included in the solution set.

Since x is less than 24, we shade the entire portion of the number line that lies to the left of the open circle. This shaded area represents all the numbers that are less than 24, and by doing so, it allows us to see at a glance all the values that x can take. When graphing inequalities, the type of circle used (open or closed) is determined by whether the variable is strictly less than (<) or less than or equal to (≤) a number, or conversely for greater than (>) and greater than or equal to (≥) symbols.

Inequality Notation

Inequality notation communicates the relationship between values and is expressed using symbols such as <, >, ≤, and ≥. These symbols stand for 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively.

In the context of the example inequality (x<24), the < symbol indicates that the variable x can take any value that is less than 24, but not 24 itself. If the inequality said (x≤24), it would include 24 as well as all numbers less than 24. Understanding and interpreting these symbols correctly is fundamental to solving and graphing inequalities, as they determine the range of possible solutions and how we represent those solutions visually.