Question

Solve the inequality. Then graph the solution set on the real number line. \(-8 \leq 1-3(x-2)<13\)

Short answer

The solution is \( 5 \geq x > -2\). In the graph, a hollow circle is drawn at -2, a solid circle at 5, and the area in between is shaded to represent the solution set.

Step-by-step solution

Simplify the equation

Firstly, simplify the central part of the inequality: \(1-3(x-2)\). Start with distributing the '-3'. Therefore:\( -8 \leq 1-(3x-6)<13 \)Simplify it to: \( -8 \leq 1-3x+6 <13 \)After combining like terms, the inequality becomes:\( -8 \leq -3x+7 <13 \)

Isolate the 'x' term

Next, isolate the 'x' term on one side of the inequality. To do this, subtract '7' from all parts of the inequality:\( -8-7 \leq -3x+7-7 <13-7 \)Which simplifies to:\( -15 \leq -3x <6 \)

Solve for 'x'

Finally, solve for 'x' by dividing all parts of the inequality by '-3'. Since we're dealing with an inequality, remember that dividing by a negative number FLIPS the direction of the inequality:\( \frac{-15}{-3} \geq \frac{-3x}{-3} > \frac{6}{-3} \)This simplifies to:\( 5 \geq x > -2 \)

Graphing the solution

The solution to the inequality is \(5 \geq x > -2\). You can graph this on a number line. Place a circle at -2 and color it hollow to represent '>' (greater than but not equal to), draw a solid circle at 5 to represent '\geq' (greater than or equal to), and shade the area in between to represent all the possible solutions to the inequality.

Key concepts

Graphing Inequalities

Graphing inequalities on a number line is a visual way of representing the solution set of an inequality. It is essential to understand how to accurately graph these solutions, as it provides a clear image of all the possible values that satisfy the inequality. In the given exercise, we are dealing with a compound inequality involving a 'greater than but not equal to' and a 'greater than or equal to' signs.

To graph the solution, which is given as the range from -2 to 5, with -2 not included but 5 included, you draw a number line. Then, mark a open circle at -2 to indicate that this number is not part of the solution set. For the value 5, you draw a closed circle to indicate that it is included in the solution set. Finally, you shade the region between the two marks to complete the graph. This shaded area represents all the values of 'x' that satisfy the original inequality. Utilizing a mixture of open and closed circles along with the shaded area helps to portray the exact nature of the inequality.

Linear Inequalities

Linear inequalities are expressions where, instead of an equals sign, we have an inequality like '>', '<', '\( \geq \)', or '\( \leq \)' between two algebraic expressions. They are similar to linear equations but instead of having a single solution, they usually have a range of solutions.

In our example, the inequality \( -8 \leq 1-3(x-2)<13 \) is simplified in steps. Simplifying the expression and isolating 'x' are the primary strategies used, just like solving a regular linear equation, except we must remember the rule about reversing the inequality sign when multiplying or dividing by a negative number. This rule is crucial as it ensures the accuracy of the solution set. After isolating 'x', we end up with a compound inequality that represents the set of all possible values of 'x' that make the original inequality true.

Inequality Notation

In the world of mathematics, expressing inequalities clearly and correctly is paramount. Inequality notation is comprised of a set of symbols used to denote the relative size of values. It includes 'greater than' (\( > \)), 'less than' (\( < \)), 'greater than or equal to' (\( \geq \)), and 'less than or equal to' (\( \leq \)).

These symbols not only indicate the relationship between numbers but also tell us whether the numbers themselves are included in a given set. For instance, \( \geq \) and \( \leq \) include the number indicated, while \( > \) and \( < \) do not. This concept comes into play when solving and graphing the inequality, wherein the direction of the inequality changes when both sides are multiplied or divided by a negative number. As shown in the example, \( 5 \geq x > -2 \) means that 'x' can be up to 5, including 5 itself, but greater than -2, not including -2.