MATH MADE EASY - MathMaster

1,344 towards social security out of their gross wages.",829,166,"employees-in-2012-paid-4-2-of-their-gross-wages-towards-social-security-while-employers-paid-another-6-2-how-much-will-someone-earning-32-000-a-year-pay-towards-social-security-out-of-their-gross-w",{"id":55,"category":47,"text_question":56,"photo_question":49,"text_answer":57,"step_text_answer":8,"step_photo_answer":8,"views":58,"likes":59,"slug":60},537924,"Write and solve an equation:\nFind a number such that 5 less than the opposite of the number is 3 times the sum of the number and -7","1. Write the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-x - 5 = 3

$x - 7$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Expand the right side: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-x - 5 = 3x - 21\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> to both sides: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-5 = 4x - 21\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Add 21 to both sides:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>16 = 4x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Divide by 4:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 4\u003C/math-field>\u003C/math-field>",1033,207,"write-and-solve-an-equation-find-a-number-such-that-5-less-than-the-opposite-of-the-number-is-3-times-the-sum-of-the-number-and-7",{"id":62,"category":47,"text_question":63,"photo_question":49,"text_answer":64,"step_text_answer":8,"step_photo_answer":8,"views":65,"likes":66,"slug":67},537879,"put brackets, addition, subtraction, multiplication or division signs to make the numerical expression correct: 6 6 6 6 6=22","Solution:\u003Cbr />\n1. We start with five 6s: \u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>6 \\ 6 \\ 6 \\ 6 \\ 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Choose appropriate operations and brackets:\u003Cbr />\n- Use addition

$+$ , subtraction

$-$ , multiplication

$*$ , and division

$/$ .\u003Cbr />\n- Insert operations between 6s and appropriate brackets to reach the result of 22.\u003Cbr />\n\u003Cbr />\n3. Apply the chosen operations and brackets to evaluate:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$6 / 6$ + 6 + 6 + 6 = 22\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Check the calculation:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>6 / 6 = 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 + 6 + 6 + 6 = 22\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. We have achieved the target result of 22.",516,103,"put-brackets-addition-subtraction-multiplication-or-division-signs-to-make-the-numerical-expression-correct-6-6-6-6-6-22",{"id":69,"category":47,"text_question":70,"photo_question":49,"text_answer":71,"step_text_answer":8,"step_photo_answer":8,"views":72,"likes":73,"slug":74},537747,"Use the definition of differentiation to differentiate \n1. f

$x$ =2x+3\n2. f

$x$ =2x^2+2x-1","1. Given \$ f(x) = 2x + 3 \$\u003Cbr />\n - Use the limit definition of a derivative: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f'(x) = \\lim_{h \\to 0} \\frac{2h}{h} \u003C/math-field>\u003C/math-field>\u003Cbr />\n - Simplify: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f'(x) = 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n Answer: \$ f'(x) = 2 \$\u003Cbr />\n\u003Cbr />\n2. Given \$ f(x) = 2x^2 + 2x - 1 \$\u003Cbr />\n - Use the limit definition of a derivative: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f'(x) = \\lim_{h \\to 0} \\frac{4xh + 2h^2 + 2h}{h} \u003C/math-field>\u003C/math-field>\u003Cbr />\n - Simplify: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f'(x) = \\lim_{h \\to 0} (4x + 2h + 2) \u003C/math-field>\u003C/math-field>\u003Cbr />\n - Substitute \$ h = 0 \$: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f'(x) = 4x + 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n Answer: \$ f'(x) = 4x + 2 \$",994,199,"use-the-definition-of-differentiation-to-differentiate-1-f-x-2x-3-2-f-x-2x-2-2x-1",{"id":76,"category":47,"text_question":77,"photo_question":49,"text_answer":78,"step_text_answer":8,"step_photo_answer":8,"views":79,"likes":80,"slug":81},537736,"Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.","1. Identify the implications of the statement \"x is an even integer if and only if 5x+3 is an odd integer.\" This is a biconditional statement, which means it can be expressed as two conditional statements (implications).\u003Cbr />\n\u003Cbr />\n2. Break down the biconditional statement into its two conditional components:\u003Cbr />\n \u003Cbr />\n - Conditional 1: If \$ x \$ is an even integer (hypothesis), then \$ 5x+3 \$ is an odd integer (conclusion).\u003Cbr />\n \u003Cbr />\n - Conditional 2: If \$ 5x+3 \$ is an odd integer (hypothesis), then \$ x \$ is an even integer (conclusion).\u003Cbr />\n\u003Cbr />\n3. Combine the two conditionals with \"and\" to form the conjunction of the conditionals:\u003Cbr />\n \u003Cbr />\n - \"If \$ x \$ is an even integer, then \$ 5x+3 \$ is an odd integer\" \$ \\land \$ \"If \$ 5x+3 \$ is an odd integer, then \$ x \$ is an even integer.\"\u003Cbr />\n\u003Cbr />\n**Answer:**\u003Cbr />\n\u003Cbr />\nIf \$ x \$ is an even integer, then \$ 5x+3 \$ is an odd integer.\u003Cbr />\n\u003Cbr />\nIf \$ 5x+3 \$ is an odd integer, then \$ x \$ is an even integer.",870,174,"suppose-x-is-an-integer-consider-the-proposition-x-is-an-even-integer-if-and-only-if-5x-3-is-an-odd-integer-express-this-proposition-as-a-conjunction-of-two-distinct-conditional-statements",{"id":83,"category":47,"text_question":84,"photo_question":49,"text_answer":85,"step_text_answer":8,"step_photo_answer":8,"views":86,"likes":87,"slug":88},537734,"Solve -2y + 6y - x - 4 = 0 for y","Solution:\u003Cbr />\n1. Given equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2y + 6y - x - 4 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Simplify the equation by combining like terms:\u003Cbr />\n * Combine terms with \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field>: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2y + 6y = 4y\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Equation becomes: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4y - x - 4 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field>:\u003Cbr />\n * Add \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field> to both sides: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4y = x + 4\u003C/math-field>\u003C/math-field>\u003Cbr />\n * Divide both sides by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field> to solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field>: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = \\frac{x + 4}{4}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThe equation solved for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = \\frac{x + 4}{4}\u003C/math-field>\u003C/math-field>.",590,118,"solve-2y-6y-x-4-0-for-y",{"id":90,"category":47,"text_question":91,"photo_question":49,"text_answer":92,"step_text_answer":8,"step_photo_answer":8,"views":93,"likes":94,"slug":95},537723,"Guido has a quarter of Andrés's money and three times as much as David's money. How much money does each have if the sum of all the amounts is $48,000?","1. Declare the variable for Andrés's money: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Determine Guido's money which is a quarter of Andrés's: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> y = \\frac{x}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Determine David's money, which is a third of Guido's: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> z = \\frac{y}{3} = \\frac{\\frac{x}{4}}{3} = \\frac{x}{12} \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Write the equation that sums their total money: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + \\frac{x}{4} + \\frac{x}{12} = 48000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Get a common denominator for the fractions: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + \\frac{3x}{12} + \\frac{x}{12} = 48000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n6. Combine the fractions: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + \\frac{4x}{12} = 48000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n7. Simplify the expression: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x + \\frac{x}{3} = 48000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n8. Eliminate the fraction by multiplying by 3: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 3x + x = 144000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n9. Simplify further: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4x = 144000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n10. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \u003C/math-field>\u003C/math-field> by dividing by 4: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 36000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nFinal amounts:\u003Cbr />\n- Andrés: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 36000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n- Guido: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> y = \\frac{36000}{4} = 9000 \u003C/math-field>\u003C/math-field>\u003Cbr />\n- David: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> z = \\frac{9000}{3} = 3000 \u003C/math-field>\u003C/math-field>",916,183,"guido-has-a-quarter-of-andres-s-money-and-three-times-as-much-as-david-s-money-how-much-money-does-each-have-if-the-sum-of-all-the-amounts-is-48-000",{"id":97,"category":47,"text_question":98,"photo_question":49,"text_answer":99,"step_text_answer":8,"step_photo_answer":8,"views":100,"likes":87,"slug":101},537715,"3a2-ab if a =-4 and b =3","1. Substitute \$ a = -4 \$ and \$ b = 3 \$ into the expression \$ 3a^2 - ab \$.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3 \\times (-4)^2 - (-4) \\times 3\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Calculate \$(-4)^2\$.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(-4)^2 = 16\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Multiply by 3.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3 \\times 16 = 48\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Calculate \$- (-4) \\times 3\$.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>- (-4) \\times 3 = 12\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Subtract the result of step 4 from the result of step 3.\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>48 + 12 = 60\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Answer: 60",589,"3a2-ab-if-a-4-and-b-3",{"id":103,"category":47,"text_question":104,"photo_question":49,"text_answer":105,"step_text_answer":8,"step_photo_answer":8,"views":106,"likes":107,"slug":108},537607,"1/12of 1=","Solution:\u003Cbr />\n1. To find \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{12}\u003C/math-field>\u003C/math-field> of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field>, multiply \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{12}\u003C/math-field>\u003C/math-field> by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. Calculation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{12} \\times 1 = \\frac{1}{12}\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nTherefore, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{12}\u003C/math-field>\u003C/math-field> of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{12}\u003C/math-field>\u003C/math-field>.",888,178,"1-12of-1",{"id":110,"category":47,"text_question":111,"photo_question":49,"text_answer":112,"step_text_answer":8,"step_photo_answer":8,"views":113,"likes":114,"slug":115},537561,"Let G be group\n If H is a subgroup of G, show that for all g∈G the set g*H*g^(-1)={g*H*g^(-1) ├|h∈H┤} is a subgroup of G\n Show that if H is finite, then g*H*g^(-1) is also finite and has the same number of elements as H.","1. **Mostrar que $gHg^{-1}$ es un subgrupo de $G$:**\u003Cbr />\n\u003Cbr />\n - **Cerradura:** Para $a, b \\in gHg^{-1}$, $a = gh_1g^{-1}$ y $b = gh_2g^{-1}$, donde $h_1, h_2 \\in H$. Entonces, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>ab = (gh_1g^{-1})(gh_2g^{-1}) = gh_1(h_2g^{-1}) = g(h_1h_2)g^{-1}.\u003C/math-field>\u003C/math-field>\u003Cbr />\n Como $H$ es subgrupo, $h_1h_2 \\in H$, entonces $ab \\in gHg^{-1}$.\u003Cbr />\n \u003Cbr />\n - **Identidad:** Como $e \\in H$ (elemento identidad de $H$) y $ge(g^{-1}) = g g^{-1} = e \\in gHg^{-1}$, la identidad de $G$ también está en $gHg^{-1}$.\u003Cbr />\n\u003Cbr />\n - **Inverso:** Para $a \\in gHg^{-1}$, $a = ghg^{-1}$, donde $h \\in H$. El inverso de $a$ es \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>a^{-1} = (ghg^{-1})^{-1} = g^{-1}(h^{-1})g.\u003C/math-field>\u003C/math-field>\u003Cbr />\n Como $h \\in H$ implica $h^{-1} \\in H$, entonces $a^{-1} = g(h^{-1})g^{-1} \\in gHg^{-1}$.\u003Cbr />\n \u003Cbr />\n2. **Mostrar que $\\#(gHg^{-1}) = \\#H$ si $H$ es finito:**\u003Cbr />\n\u003Cbr />\n - Dado que la función $f: H \\to gHg^{-1}$ tal que $f(h) = ghg^{-1}$ es un bijectivo, el conjunto $gHg^{-1}$ tiene la misma cardinalidad que $H$. \u003Cbr />\n\u003Cbr />\n Entonces, $gHg^{-1}$ es un subgrupo de $G$. Y además, si $H$ es finito, $gHg^{-1}$ es finito y tiene el mismo número de elementos que $H$. Así, $\\#(gHg^{-1}) = \\#H$.",1151,230,"let-g-be-group-if-h-is-a-subgroup-of-g-show-that-for-all-g-g-the-set-g-h-g-1-g-h-g-1-h-h-is-a-subgroup-of-g-show-that-if-h-is-finite-then-g-h-g-1-is-also-finite-and-has-the-same-numb",{"first":6,"last":117,"prev":8,"next":10},12,{"current_page":6,"from":6,"last_page":117,"links":119,"path":153,"per_page":144,"to":144,"total":42},[120,123,126,128,130,132,134,137,140,143,146,149,151],{"url":6,"label":121,"active":122},"1",true,{"url":10,"label":124,"active":125},"2",false,{"url":13,"label":127,"active":125},"3",{"url":16,"label":129,"active":125},"4",{"url":19,"label":131,"active":125},"5",{"url":22,"label":133,"active":125},"6",{"url":135,"label":136,"active":125},7,"7",{"url":138,"label":139,"active":125},8,"8",{"url":141,"label":142,"active":125},9,"9",{"url":144,"label":145,"active":125},10,"10",{"url":147,"label":148,"active":125},11,"11",{"url":117,"label":150,"active":125},"12",{"url":10,"label":152,"active":125},"Next 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