Review Exercise 5 — Class 9 Mathematics Solutions

Solution

Solution

1. Understand the definition of a linear equation.

   A linear equation in one variable $x$ is an algebraic statement of the form:

   $$\begin{array}{cc}ax+b& =0\end{array}$$

   where $a,b\in ℝ$ with $a\ne 0$. It satisfies two conditions:

   1. It contains an equality relation $(=)$.
   2. The highest exponent (degree) of the variable $x$ is $1$.

2. Evaluate each given option.

   • Option (a): $5x>7$ contains a strict inequality sign $(>)$, so it is a linear inequality, not an equation.
   • Option (b): $4x-2<1$ contains a strict inequality sign $(<)$, making it a linear inequality.
   • Option (c): $2x+1=1$ contains an equality sign $(=)$ and the variable $x$ has an exponent of $1$ with a non-zero coefficient $2$. Therefore, it is a linear equation.
   • Option (d): $4=1+3$ contains an equality sign, but it contains no variable; it is purely a numerical identity, not an algebraic linear equation.

3. Conclusion.

   Option (c) is the only linear equation.

Solution

1. Isolate the variable term.

   Given the linear equation:

   $$\begin{array}{cc}5x-10& =10\end{array}$$

   Add $10$ to both sides:

   $$\begin{array}{cc}5x& =10+10\\ 5x& =20\end{array}$$

2. Solve for $x$.

   Divide both sides by $5$:

   $$\begin{array}{cc}x& =\frac{20}{5}\\ x& =4\end{array}$$

3. Verify and analyze distractors.

   Substitute $x=4$ into the original equation:

   $$\begin{array}{cc}5(4)-10& =20-10\\ & =10\end{array}$$

   The left-hand side matches the right-hand side.

   • Option (a) $0$: Occurs if one incorrectly subtracts $10-10=0$.
   • Option (b) $50$: Occurs if one incorrectly multiplies $10\times 5$.
   • Option (c) $4$: Correct value.
   • Option (d) $-4$: Result of a sign error.

Solution

1. Solve the linear inequality.

   Given:

   $$\begin{array}{cc}7x+4& <6x+6\end{array}$$

   Subtract $6x$ from both sides:

   $$\begin{array}{cc}7x-6x+4& <6\\ x+4& <6\end{array}$$

   Subtract $4$ from both sides:

   $$\begin{array}{cc}x& <6-4\\ x& <2\end{array}$$

2. Express the solution in interval notation.

   The inequality $x<2$ represents all real numbers strictly less than $2$. Since the inequality is strict $(<)$, the endpoint $2$ is not included (represented by a round parenthesis):

   $$\begin{array}{cc}x& \in (-\infty ,2)\end{array}$$

3. Analyze distractors.

   • Option (a) $(2,\infty )$: Represents $x>2$, reversing the inequality.
   • Option (b) $[2,\infty )$: Represents $x\ge 2$, reversing direction and improperly including the endpoint.
   • Option (c) $(-\infty ,2)$: Correct strict interval.
   • Option (d) $(-\infty ,2]$: Incorrectly includes the boundary point $2$ with a square bracket $(x\le 2)$.

Solution

1. Understand plane partitioning by a line.

   In coordinate geometry, any straight line $L$ drawn on the Cartesian plane partitions the points of the plane into three disjoint sets:

   1. Points lying on the line $L$ itself.
   2. Points lying on one side of $L$.
   3. Points lying on the opposite side of $L$.

   The regions on either side are each called a half-plane. Together, the line creates two half planes.

2. Consider the specific case of a vertical line.

   A vertical line has equation:

   $$\begin{array}{cc}x& =c\end{array}$$

   It divides the Cartesian plane into:

   1. The left half plane, defined by $x<c$.
   2. The right half plane, defined by $x>c$.

   Thus, the vertical line divides the plane into two half planes (the left half plane and the right half plane).

3. Analyze distractors.

   • Options (a) and (b): Each names only one of the two regions resulting from the division.
   • Option (c): Incorrect because a line partitions the plane rather than keeping it whole.
   • Option (d): Correct because the division results in two half planes.

Solution

1. State the definition.

   In the study of linear inequalities and linear programming: When the inequality symbol $(<,>,\le ,\text{ or }\ge )$ in a linear inequality such as:

   $$\begin{array}{cc}ax+by& <c\text{or}ax+by\le c\end{array}$$

   is replaced by an equality symbol $(=)$, the resulting equation:

   $$\begin{array}{cc}ax+by& =c\end{array}$$

   is specifically defined as the associated equation (also known as the boundary equation or corresponding equation).

2. Purpose of the associated equation.

   The graph of the associated equation is a straight line that acts as the boundary separating the plane into two half-planes.

3. Analyze distractors.

   • Option (a): While the resulting equation is linear, the specific terminology used in the curriculum for the equation derived from an inequality is the "associated equation".
   • Option (b): Exactly matches the formal textbook definition.
   • Option (c): Incorrect because the degree of the variables remains $1$, not $2$.
   • Option (d): Incorrect since option (b) is the correct term.

Solution

1. Identify the components of the expression.

   The given mathematical statement is:

   $$\begin{array}{cc}3x+4& <0\end{array}$$

   It consists of an algebraic expression $3x+4$ and a constant $0$ connected by the strictly less than relational operator $(<)$.

2. Apply mathematical definitions.

   • An inequality is any mathematical statement stating that one expression is greater than, greater than or equal to, less than, or less than or equal to another expression using symbols $>,\ge ,<,\le$.
   • An equation asserts that two expressions are equal using $(=)$.
   • An identity is an equation that holds true for all permissible values of the variable.

   Since $3x+4<0$ contains the symbol $(<)$, it is an inequality.

3. Analyze distractors.

   • Option (a) equation: Incorrect because it lacks the equality symbol $(=)$.
   • Option (b) inequality: Correct formal classification.
   • Option (c) not inequality: False statement.
   • Option (d) identity: Incorrect because it is neither an equation nor universally true for all $x\in ℝ$.

Solution

1. Define corner point in linear programming.

   In linear inequalities and linear programming: A corner point of a feasible region is a point in the solution space where two boundary lines intersect.

2. Geometric correspondence.

   In geometry:

   • A point where two straight line segments or boundary lines meet to form an angle or corner of a polygon is called a vertex (plural: vertices).
   • Therefore, in linear programming literature and geometry, a corner point of the solution or feasible region is also called a vertex.

3. Analyze distractors.

   • Option (a) code: Unrelated mathematical or computing term.
   • Option (b) vertex: Correct standard mathematical synonym.
   • Option (c) curve: A continuous one-dimensional geometric object, not an isolated point.
   • Option (d) region: A two-dimensional set of points or area, not an intersection point.

Solution

1. Understand the test point method.

   A point $(x_0,y_0)=(0,0)$ is a solution to an inequality if substituting $x=0$ and $y=0$ into the inequality results in a true mathematical statement.

2. Test each option at $(0,0)$.

   • Option (a): $4x+5y>8$

   $$\begin{array}{cc}4(0)+5(0)& >8\\ 0+0& >8\\ 0& >8(\text{False})\end{array}$$

   • Option (b): $3x+y>6$

   $$\begin{array}{cc}3(0)+0& >6\\ 0+0& >6\\ 0& >6(\text{False})\end{array}$$

   • Option (c): $-2x+3y<6$

   $$\begin{array}{cc}-2(0)+3(0)& <6\\ 0+0& <6\\ 0& <6(\text{True})\end{array}$$

   • Option (d): $x+y>4$

   $$\begin{array}{cc}0+0& >4\\ 0& >4(\text{False})\end{array}$$

3. Conclusion.

   Since $0<6$ is true, the origin $(0,0)$ satisfies only the inequality in option (c).

Solution

1. Understand constraints and non-negativity.

   In a linear programming problem (LPP), the system contains:

   1. Problem constraints (linear inequalities describing physical/economic limitations).
   2. Non-negativity constraints:

   $$\begin{array}{cc}x& \ge 0\\ y& \ge 0\end{array}$$

   These non-negativity constraints restrict all valid solutions strictly to the first quadrant (including its boundary axes).

2. Define feasible region.

   The set of all points satisfying all the given system of linear inequalities together with the non-negativity constraints $(x\ge 0,y\ge 0)$ is termed the feasible region. Each point within this region is called a feasible solution.

3. Analyze distractors.

   • Option (a) objective region: Fictitious terminology; the objective function is the linear expression to be optimized, not a region.
   • Option (b) feasible region: Correct standard technical term.
   • Option (c) solution region: A general term for the graph of any inequality or system of inequalities without necessarily imposing the first-quadrant non-negativity condition.
   • Option (d) constraints region: Non-standard descriptive phrase, not the formal mathematical term.

Solution

1. Define optimization in linear programming.

   In linear programming: The primary goal is to find optimal decisions (values of variables $x$ and $y$) that maximize a benefit (such as profit or revenue) or minimize a cost (such as expenditure or loss).

2. Formal definition of the objective function.

   The linear function of the decision variables:

   $$\begin{array}{cc}z& =f(x,y)\\ & =ax+by\end{array}$$

   which is formulated to be maximized or minimized subject to a set of linear constraints is termed the objective function.

3. Analyze distractors.

   • Option (a) solution function: Incorrect, not a standard mathematical term in optimization.
   • Option (b) objective function: Exactly matches the formal definition.
   • Option (c) feasible function: Incorrect terminology.
   • Option (d) none of these: Incorrect since option (b) is the correct term.

Solution

Quick Solution

Under Review

Quick Solution

Under Review

Quick Solution

Under Review

Quick Solution

Under Review

Solution

Quick Solution

Under Review

Quick Solution

Under Review

Quick Solution

Under Review

Quick Solution

Under Review