Question

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x>-2 $$

Short answer

Yes, the point \((0,0)\) is a solution to the inequality \(x > -2\). The graph is a number line with shading to the right from \(-2\).

Step-by-step solution

Checking if the Point is a Solution

Replace \(x\) with \(0\) in the inequality \(x > -2\). This gives \(0 > -2\), which is a true statement, so \((0,0)\) is a solution to the inequality.

Graphing the Inequality

For \(x > -2\), we will draw a line on the number line at \(-2\) and shade to the right of \(-2\) since \(x\) is greater than \(-2\). As \((0,0)\) is a solution to the inequality, it will be included in the shaded region.

Key concepts

Inequality Solutions

When dealing with algebraic inequalities, we often seek to determine the set of all possible values that satisfy an inequality. These sets are known as inequality solutions. For instance, consider the inequality \(x > -2\). To find its solutions, we look for all the numbers that are greater than -2. In our example, testing whether point \((0,0)\) invalidates or validates this inequality is crucial, as it helps us understand whether this specific value is part of the solution set.

It's important to note that solutions to inequalities are not just single numbers but entire ranges. Since \(0 > -2\) is a true statement, \((0,0)\) is indeed a solution to the inequality. This tells us that all values greater than -2 make the inequality true, thus creating an infinite set of solutions to the right of -2 on the number line.

Number Line Graphing

Graphing on a number line is a visual way to represent inequality solutions. For an inequality like \(x > -2\), you would depict this on a number line by first drawing a dot or a circle at -2. Since the inequality is strict (\(>\text{, not }\geq\)), we use an open circle to indicate that -2 is not included in the solution set. After placing the open circle, we then shade or draw an arrow to the right, signifying all the numbers that are greater than -2.

Tip for Clarity

Always use a clear and distinct mark to represent the pivotal number in the inequality. Using different colors or shading styles can help distinguish between the included and excluded values. Moreover, labeling the number line with significant values ensures that anyone who views the graph can easily understand where the solutions lie.

Algebraic Inequalities

Algebraic inequalities are statements that describe a range of values that a variable can assume. These inequalities are represented by symbols such as \(>\), \(<\), \(\geq\), and \(\leq\). Each symbol has a specific meaning and indicates whether the range of solutions includes the boundary number. In the case of strict inequalities, such as \(x > -2\), the variable \(x\) can take any value greater than -2, but not -2 itself.

Boundary Values and Strictness

The term 'strict' refers to the absence of the equal sign in the inequality, which means the solutions approach but do not include the boundary number. Understanding 'strictness' in inequalities helps you decide how to graph the inequality on a number line and whether to use an open or closed dot to represent the boundary value.