# Solutions and Explanations to Intermediate Algebra Questions in Sample 2

Solutions and full explanations of intermediate algebra questions in sample 2 are presented.

 (True or False)     The inequality |x + 1| < 0 has no solution. Solution The absolute value of a real expression is either positive or equal to zero. Therefore there is no value of x that makes |x + 1| negative and therefore |x + 1| < 0 is never true and the statement "The inequality |x + 1| < 0 has no solution" is TRUE. (True or False)     If a and b are negative numbers, and |a| < |b|, then b - a is negative. Solution Since a and b are both negative, they are positioned to the left of zero on the number line. Since |a| < |b|, a is closer to zero than b and therefore a is greater than b which written as a > b Subtract a to both sides and simplify a - a > b - a 0 > b - a Hence the statement "b - a is negative" is TRUE. (True or False)     The equation 2x + 7 = 2(x + 5) has one solution. Solution Let us solve the given equation 2x + 7 = 2(x + 5) 2x + 7 = 2x + 10 , expand right hand term 2x + 7 - 2x = 2x + 10 - 2x , subtract 2x from both sides 7 = 10 , simplify The above statement is never true and therefore the given equation has no solutions. The statement "The equation 2x + 7 = 2(x + 5) has one solution" is FALSE. (True or False)     The multiplicative inverse of -1/4 is -1/8. Solution If a real number x is not equal to zero, its multiplicative inverse is equal to 1/x. Hence the inverse of -1/4 is equal to 1 / (-1/4) = (1/1) / (-1/4) = (1/1)*(-4/1) = - 4 and therefore the statement "The multiplicative inverse of -1/4 is -1/8" is FALSE. (True or False)     x ÷ (2 + z) = x ÷ 2 + x ÷ z Solution let us use the values x = 8 and z = 2 and evaluate the values of the left side and right side expressions. Left side: x ÷ (2 + z) = 8 ÷ (2 + 2) = 2 Right side: x ÷ 2 + x ÷ z = 8 ÷ 2 + 8 ÷ 2 = 4 + 4 = 8 Since x ÷ (2 + z) = x ÷ 2 + x ÷ z is not true for one value of x and one value of z, the statement is FALSE. (True or False)     |-8| - |10| = -18 Solution Evaluate left side. |-8| - |10| = 8 - 10 = - 2 hence the statement |-8| - |10| = -18 is FALSE. (True or False)     (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2) Solution Evaluate left side. (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1 Evaluate right side. 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4 hence the statement (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2) is FALSE. (True or False)     31.5(1.004)20 < 31.6(1.003)25 Solution Use calculator and calculate left and right sides of inequality. 31.5(1.004)20 = 34.118 (rounded to 3 decimal places) 31.6(1.003)25 = 34.057 (rounded to 3 decimal places) since 34.118 is greater than 34.057 the statement 31.5(1.004)20 < 31.6(1.003)25 is FALSE. (True or False)     The graph of the equation y = 4 has no x-intercept. Solution The line with equation y = 4 is a horizontal line parallel to the x axis and hence cannot intersect the x axis. The statement " the graph of the equation y = 4 has no x-intercept" is TRUE. (True or False)     The value of n(n + 3)/2 = 3/2 when n = 0. Solution Evaluate n(n + 3)/2 = 3/2 for n = 0. n(n + 3)/2 = 0(0 + 3) / 2 = 0 / 2 = 0 The statement "the value of n(n + 3)/2 = 3/2 when n = 0" is FALSE. (True or False)     The distance between the numbers -9 and 20 is equal to the distance between 9 and -20 on the number line. Solution Distance between numbers a and b on a number line is given by . |a - b| = |b - a| Hence, distance between -9 and 20 is equal to |-9 - 20| = 29 Hence, distance between 9 and - 20 is equal to |9 - (-20)| = 29 The two distances are equal. (True or False)     If f(x) = √(1 - x), then f(-3) = 2. Solution f(-3) = √(1 - (-3)) = √4 = 2 (True or False)     The slope of the line 2x + 2y = 2 is equal to 2. Solution Write equation in slope intercept y = m x + b form and identify its slope m. Given equation 2x + 2y = 2 Subtract 2 x from both sides and simplify 2 y = - 2x + 20 Divide all terms by 2 y = - x + 10 Slope is equal to - 1 (True or False)     |x + 5| is always positive. (True or False)     The distance between the points (0 , 0) and (5 , 0) in a rectangular system of axes is 5. (True or False)     1 / (2x - 4) is undefined when x = -4. (True or False)     (-1/5)-2 = 25. (True or False)     The reciprocal of 0 is equal to 0. (True or False)     The additive inverse of -10 is equal to 10. (True or False)     1 / (x - 4) = 1/x - 1/4. Answers to the Above Questions TRUE TRUE FALSE (2x + 7 = 2x + 10 , 7 = 10 no solution) FALSE ( multiplicative inverse of -1/4 is -4) FALSE (try the values x = 8 and z = 2) FALSE ( = 8 - 10 = -2) FALSE (left side = 1 , right side = 4) FALSE TRUE FALSE ( =0 ) TRUE TRUE FALSE ( = -1 ) FALSE ( = 0 for x = -5) TRUE FALSE (undefined when 2x - 4 = 0 , x = 2) TRUE FALSE (the reciprocal of 0 is undefined) TRUE FALSE (try x = 2) Algebra Questions and problems More ACT, SAT and Compass practice