Solve Algebra and Linear Programming Problems with Inequalities and Intercepts

Math Problem Statement

Which of the following is NOT true? A. (x - 4)^2 = (4 - x)^2 B. If -3(x - 4) < (14 -x) then (x - 4) < -3 (x - 14) C. None of the other options D. If (x - 4) < 3 then x > 7 When a linear programming problem is about a factory manufacturing two types of washing machines X and Y, then one of the implicit constraints has to be : A. X, Y are Integers B. X, Y are Natural numbers C. X, Y are Whole numbers D. X, Y are Real numbers When a linear programming problem is about two pumps X and Y which supply water to a reservoir, each at a specific liters per hour rate, then one of the implicit constraints has to be : A. X, Y are Natural numbers B. X, Y are Real numbers C. X, Y are Integers D. X, Y are Whole numbers The coordinates of the intercepts with the X- and Y-axes of 3y - 2x - 6 = 6 are: A. (4;0) and (0;-6) B. (0;0) and (-6;4) C. (-6;0) and (0;4) D. (-6;4) and (4;-6) The origin with coordinates (0;0) falls in the feasible region of 3y - 2x - 6 ≤ 6 True False

Solution

Let's go through each question step by step.

1. Which of the following is NOT true?

A. $(x - 4)^2 = (4 - x)^2$

This is true because $(x - 4)$ and $(4 - x)$ are negatives of each other, and squaring a number always makes it positive: $(x - 4)^2 = (-(x - 4))^2 = (4 - x)^2.$

B. If $-3(x - 4) < (14 - x)$ , then $(x - 4) < -3(x - 14)$

Let's solve this inequality:

$-3(x - 4) < (14 - x) \implies -3x + 12 < 14 - x.$ Add $3x$ and $x$ to both sides: $12 + 2x < 14 \implies 2x < 2 \implies x < 1.$

Now, testing the second inequality $(x - 4) < -3(x - 14)$ : $x - 4 < -3(x - 14) \implies x - 4 < -3x + 42.$ Add $3x$ to both sides: $4x - 4 < 42 \implies 4x < 46 \implies x < 11.5.$

Clearly, $x < 1$ is not equivalent to $x < 11.5$ , so this is false.

C. None of the other options

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Programming
Inequalities
Coordinate Geometry

Formulas

(x - a)^2 = (a - x)^2
Linear equations in two variables (Ax + By + C = 0)

Theorems

Properties of quadratic and linear equations
Basic properties of inequalities
Feasibility region in linear programming

Suitable Grade Level

Grades 9-12

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