what are system of linear inequalities?
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A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system. Example.
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1 Expert Answer A linear inequality is considering one value where as a compound inequality has two values to be considered in the situation to be determined if the inequlaity(ies) are true or not.
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Linear Combination of Vectors. If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors. For example, suppose a = 2b + 3c, as shown below. ... Thus, a is a linear combination of b and c.
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In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contains the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S.
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['Factor the polynomials: ... ', 'Make the coefficient of the variable of all factors positive: ... ', 'Multiply/divide both sides of the inequality by -1 to remove the minus sign (remember that in doing so the inequality would reverse): ... ', 'Find the roots and asymptotes of the inequality by equating each factor to 0:']
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['Binary Step Function. ... ', 'Linear Function. ... ', 'Sigmoid. ... ', 'Tanh. ... ', 'ReLU. ... ', 'Leaky ReLU. ... ', 'Parameterised ReLU. ... ', 'Exponential Linear Unit.']
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A system of linear inequalities is often used to determine the maximum or minimum values of a situation with multiple constraints. For example, you might be determining how many of a product should be produced to maximize a profit.
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Inequalities are the relationships between two expressions which are not equal to one another. The symbols used for inequalities are <, >, ≤, ≥ and ≠. reads as '7 is greater than ' (or ' is less than 7', reading from right to left).
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If, on the other hand, there exists a nontrivial linear combination that gives the zero vector, then the vectors are dependent. Example 2: Use this second definition to show that the vectors from Example 1— v 1 = (2, 5, 3), v 2 = (1, 1, 1), and v 3 = (4, −2, 0)—are linearly independent.
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There is another method, called the test point method, that can be used instead of the sign graph to solve inequalities. The following example shows how to solve a quadratic inequality that involves a prime polynomial. Solve x2 - 4x - 6 > 0 and graph the solution set.
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We can solve quadratic inequalities graphically by first rewriting the inequality in standard form, with zero on one side. Graph the quadratic function and determine where it is above or below the x-axis. If the inequality involves “less than,” then determine the x-values where the function is below the x-axis.
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The following results from Section 1.7 are still true for more general vectors spaces. A set containing the zero vector is linearly dependent. A set of two vectors is linearly dependent if and only if one is a multiple of the other. A set containing the zero vector is linearly independent.
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['Lower bound: 7.94. Upper bound: 23.66.', 'Lower bound: 8.59. Upper bound: 20.63. ... ', 'Lower bound: 6.61. Upper bound: 31.36.']
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The columns of matrix A are linearly independent if and only if the equation Ax 0 has only the trivial solution. Special Cases. Sometimes we can determine linear independence of a set with minimal effort.
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Systems of equations that have the same solution are called equivalent systems.
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We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.
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The graph of linear inequalities include a dashed line if they are greater than or less than but not equal to. Linear equations, on the other hand, include a solid line in every situation. Moreover, linear inequalities include shaded regions whereas linear equations do not.