shaded
Graph it (the equation).
graph
When we plot all the points that satisfy an equation or inequality, we create a graphical representation of the solution set. For equations, this typically results in a curve or line that represents all combinations of variables that make the equation true. For inequalities, the plot may include shaded regions or half-planes that indicate all points meeting the condition. This visual representation helps to easily identify the solutions and their relationships in the coordinate system.
Yes, and no. The solution set to an inequality are those points which satisfy the inequality. A linear inequality is one in which no variable has a power greater than 1. Only if there are two variables will the solution be points in a plane; if there are more than two variables then the solution set will be points in a higher space, for example the solution set to the linear inequality x + y + z < 1 is a set of points in three dimensional space.
Algebraically, solutions to an equation yield specific values that satisfy the equality, while solutions to an inequality provide a range of values that satisfy the condition (e.g., greater than or less than). Graphically, an equation is represented by a distinct curve or line where points satisfy the equality, whereas an inequality is represented by a shaded region that indicates all points satisfying the inequality, often including a boundary line that can be either solid (for ≤ or ≥) or dashed (for < or >). This distinction highlights the difference in the nature of solutions: precise for equations and broad for inequalities.
graph
graph
Graph it (the equation).
graph
The shaded region above or below the line in the graph of a linear inequality is called the solution region. This region represents all the possible values that satisfy the inequality. Points within the shaded region are solutions to the inequality, while points outside the shaded region are not solutions.
It can represent the graph of a strict inequality where the inequality is satisfied by the area on one side of the dashed line and not on the other. Points on the line do not satisfy the inequality.
When we plot all the points that satisfy an equation or inequality, we create a graphical representation of the solution set. For equations, this typically results in a curve or line that represents all combinations of variables that make the equation true. For inequalities, the plot may include shaded regions or half-planes that indicate all points meeting the condition. This visual representation helps to easily identify the solutions and their relationships in the coordinate system.
Yes, and no. The solution set to an inequality are those points which satisfy the inequality. A linear inequality is one in which no variable has a power greater than 1. Only if there are two variables will the solution be points in a plane; if there are more than two variables then the solution set will be points in a higher space, for example the solution set to the linear inequality x + y + z < 1 is a set of points in three dimensional space.
We identify a set of points in the relevant space which are part of the solution set of the equation or inequality. The space may have any number of dimensions, the solution set may be contiguous or in discrete "blobs".
A dotted line in a graph of an inequality indicates that the boundary line is not included in the solution set. This typically occurs with inequalities using "<" or ">", meaning that points on the dotted line do not satisfy the inequality. In contrast, a solid line would indicate that points on the line are included in the solution set, as seen with "<=" or ">=".
To solve the inequality (8x^2 - x < 0), we first factor it as (x(8x - 1) < 0). The critical points are (x = 0) and (x = \frac{1}{8}). Analyzing the sign of the product in the intervals determined by these points, we find that the inequality holds for (0 < x < \frac{1}{8}). Since there are no integer values of (x) in this interval, the number of different integer values of (x) that satisfy the inequality is zero.
Take a sample point from either the top or bottom of the graph. I like to use (0,0) if it is not on the line. Substitute it into the inequality and if it is true then it represents all points on that line as true and vice versa.