Question

Solve the inequality and graph the solution. Then check graphically whether the given \(x\) -value is a solution by graphing the \(x\) -value on the same number line. $$ 7 \leq-2 x+21<31 ; x=0 $$

Short answer

The solution to the inequality \(7 \leq-2 x+21<31\) is \(-5 < x \leq 7\). \(x = 0\) is a valid solution to the inequality as it lies within the solution range. The graph will show a line between and including -5 and 7 with a marked point at 0.

Step-by-step solution

Solve the inequality

To solve the inequality, first subtract 21 from all three sides to isolate the \(x\)-terms. This gives \(-14 \leq -2x < 10\). Then divide all sides by -2. Remember that when multiplying or dividing by a negative number, the direction of the inequality sign switches. So, we will get \(7 \geq x > -5\). But typically we write inequalities in the format 'less than', so rewriting gives us \(-5 < x \leq 7\). This tells us that x is greater than -5 but at the same or less than 7.

Check if \(x = 0\) is a solution

Now check whether \(x = 0\) is a solution to the inequality. Substitute \(x = 0\) into the inequality \(-5 < x \leq 7\). The inequality becomes \(-5 < 0 \leq 7\), which is true. Therefore, \(x = 0\) is a solution to the inequality.

Graph the solution

To graph the solution on a number line, first draw a number line and mark the points -5 and 7. Because \(x > -5\), draw a open circle at -5 (indicating not included). Because \(x \leq 7\), draw a closed circle at 7 (indicating that it is included). Draw a line connecting the two points. To represent the solution \(x = 0\), draw a point at \(x = 0\). This point also falls within the line of our solution range, confirming that \(x = 0\) is a solution.

Key concepts

Inequality Graphing

When representing inequalities visually, inequality graphing on a number line is an intuitive way to understand the range of solutions. For inequalities involving a single variable, like the example with the inequality \(7 \leq -2x + 21 < 31; x = 0\), the first step is isolating the variable on one side. In this case, it was achieved by subtracting 21 from each part of the compound inequality, and then dividing by -2, while flipping the inequality signs.

Graphing the resulting inequality \( -5 < x \leq 7\) involves marking critical values on a number line: an open circle at -5 signifies that this number is not part of the solution, and a closed circle at 7 includes this endpoint in the solution set. Linking these points with a line or arrow visually represents all possible x-values within the solution range. A point at \(x = 0\) on the graph, within this interval, confirms graphically that it's part of the solution.

Number Line

The number line is a fundamental visual tool in mathematics that represents numbers spread along a horizontal line. It is particularly useful for displaying solutions to inequalities. To properly define the solution set for inequalities on a number line, such as \( -5 < x \leq 7\), you start by plotting the extremities -5 and 7.

An important thing to note is the distinction between open and closed circles: open circles signify that the endpoint is not included in the solution set, while closed circles mean the value is included. This distinction is crucial in accurately representing the set of all numbers that satisfy the inequality. The line between the circles indicates the subset of numbers on the number line that are part of the solution.

Negative Number Operations

Operating with negative numbers can be tricky, and it's crucial when solving inequalities. In particular, when you multiply or divide an inequality by a negative number, you must flip the direction of the inequality sign to maintain a true statement.

This concept is seen in our example where dividing by -2 across \( -14 \leq -2x < 10\) necessitates reversing the inequality signs, resulting in \(7 \geq x > -5\). Understanding these operations is essential for correctly solving inequalities and preventing common mistakes that could lead to incorrect solutions.

Substitution Method

The substitution method is a technique used to test if a certain value is a solution to an inequality. It involves replacing the variable with a given value and checking whether the result is a true statement. As seen in the exercise, \(x=0\) is substituted into the inequality \( -5 < x \leq 7\) to get \( -5 < 0 \leq 7\), which confirms that zero is in fact a solution.

This method not only verifies potential solutions but also aids in the understanding of the inequality's solution set. It's a straightforward yet powerful tool that can be applied across a range of mathematical concepts.