[{"_1":2,"_3":-5,"_4":-5},"loaderData",{"_5":6,"_7":8,"_9":-5,"_10":11,"_12":13,"_14":15,"_16":17,"_18":19},"actionData","errors","root",{"_7987":7988,"_7989":7990,"_7991":7992,"_7993":-5,"_7994":47,"_7995":7996,"_7997":47,"_7998":47,"_7999":8000,"_8001":25,"_8002":8003,"_8004":47,"_8005":8006,"_8007":-5,"_8008":-5,"_8009":-5,"_8010":-5,"_8011":-5,"_8012":-5,"_8013":8014,"_8015":8016,"_8017":25,"_8018":-5,"_24":25},"routes/__learning-environment",{"_559":1894,"_1895":1896,"_1897":1898,"_1899":1900,"_24":25},"routes/__learning-environment/course","routes/__learning-environment/course/$courseSlug",{"_526":527,"_528":529,"_530":25,"_24":25},"routes/__learning-environment/course/$courseSlug/books",{"_525":47,"_24":25},"routes/__learning-environment/course/$courseSlug/books/$methodSlug.$bookSlug",{"_114":115,"_116":117,"_118":47,"_119":120,"_121":25,"_24":25},"routes/__learning-environment/course/$courseSlug/books/$methodSlug.$bookSlug/chapters.$chapterSlug/sections.$sectionSlug",{"_108":109,"_110":111,"_112":113,"_24":25},"routes/__learning-environment/course/$courseSlug/books/$methodSlug.$bookSlug/chapters.$chapterSlug/sections.$sectionSlug/faq",{"_20":21,"_22":23,"_24":25},"backButtonHref","/course/wiskunde/books/kern/wiskunde-3-gymnasiumvwo/chapters/5-vergelijkingen-en-ongelijkheden/sections/3-ongelijkheden","questions",[26,27,28,29,30,31,32,33],"success",true,{"_34":100,"_36":101,"_38":102,"_40":103,"_42":104,"_44":105,"_46":47,"_48":25,"_49":50,"_51":50,"_52":106,"_54":107},{"_34":93,"_36":94,"_38":95,"_40":96,"_42":95,"_44":97,"_46":47,"_48":25,"_49":50,"_51":50,"_52":98,"_54":99},{"_34":85,"_36":86,"_38":87,"_40":88,"_42":89,"_44":90,"_46":47,"_48":25,"_49":50,"_51":50,"_52":91,"_54":92},{"_34":78,"_36":79,"_38":80,"_40":81,"_42":80,"_44":82,"_46":47,"_48":25,"_49":50,"_51":50,"_52":83,"_54":84},{"_34":70,"_36":71,"_38":72,"_40":73,"_42":74,"_44":75,"_46":47,"_48":25,"_49":50,"_51":50,"_52":76,"_54":77},{"_34":63,"_36":64,"_38":65,"_40":66,"_42":65,"_44":67,"_46":47,"_48":25,"_49":50,"_51":50,"_52":68,"_54":69},{"_34":56,"_36":57,"_38":58,"_40":59,"_42":58,"_44":60,"_46":47,"_48":25,"_49":50,"_51":50,"_52":61,"_54":62},{"_34":35,"_36":37,"_38":39,"_40":41,"_42":43,"_44":45,"_46":47,"_48":25,"_49":50,"_51":50,"_52":53,"_54":55},"uuid","fc32c57d-825b-4895-8f8f-a018661d9972","slug","hoe-los-ik-lineaire-ongelijkheden-op","question","Hoe los ik lineaire ongelijkheden op?","cleanedQuestion","los lineaire ongelijkheden","detailedQuestion","Hoe los ik lineaire ongelijkheden op en waar moet ik op letten bij het ongelijkheidsteken?","answer","Lineaire ongelijkheden los je op op een manier die erg lijkt op het oplossen van lineaire vergelijkingen. Het doel is om de waarde van de onbekende variabele (meestal $x$) te vinden die voldoet aan de ongelijkheid. Het belangrijkste verschil is dat je extra goed moet opletten wanneer je deelt of vermenigvuldigt met een negatief getal.\n\nDe stappen voor het oplossen van lineaire ongelijkheden zijn:\n\n1. **Breng alle termen met de variabele naar één kant en alle constante getallen naar de andere kant.**\n Dit doe je door aan beide kanten van het ongelijkheidsteken dezelfde bewerking uit te voeren (optellen of aftrekken).\n\n2. **Isoleer de variabele.**\n Dit doe je door te delen of te vermenigvuldigen met de coëfficiënt van de variabele.\n\n3. **Let op het ongelijkheidsteken bij vermenigvuldigen of delen door een negatief getal.**\n Dit is de meest cruciale stap. Als je beide kanten van een ongelijkheid vermenigvuldigt of deelt door een **negatief getal**, moet je het **ongelijkheidsteken omdraaien**.\n * `>` wordt `<`\n * `<` wordt `>`\n * `≥` wordt `≤`\n * `≤` wordt `≥`\n\n**Voorbeeld:**\nLaten we de ongelijkheid $5x - 7 \\le 11x + 17$ oplossen.\n\n1. **Breng x-termen naar één kant en constanten naar de andere:**\n We brengen $11x$ naar de linkerkant en $-7$ naar de rechterkant.\n $5x - 11x \\le 17 + 7$\n $-6x \\le 24$\n\n2. **Isoleer de variabele $x$ en let op het ongelijkheidsteken:**\n We moeten nu delen door $-6$. Omdat $-6$ een negatief getal is, moeten we het ongelijkheidsteken omdraaien.\n $\\frac{-6x}{-6} \\ge \\frac{24}{-6}$\n $x \\ge -4$\n\nDe oplossing van de ongelijkheid $5x - 7 \\le 11x + 17$ is dus $x \\ge -4$. Dit betekent dat elke waarde van $x$ die groter is dan of gelijk is aan $-4$ voldoet aan de ongelijkheid.","isConcept",false,"isSynced","likeCount",0,"dislikeCount","createdAt",["D",1777072703391],"updatedAt",["D",1777072703643],"0f22fd95-6f69-4ff4-ac22-c04e7d8b3bdc","constante-factor-nulpunten-kwadratische-vergelijking","Heeft een constante factor invloed op de nulpunten van een kwadratische vergelijking?","constante factor invloed nulpunten kwadratische vergelijking","Nee, een constante factor heeft geen invloed op de nulpunten (of wortels) van een kwadratische vergelijking. De nulpunten zijn de waarden van $x$ waarvoor de vergelijking gelijk is aan nul, oftewel de snijpunten met de x-as.\n\nWanneer een kwadratische vergelijking al is ontbonden in factoren, zoals in de vorm $a(x-p)(x-q)=0$, dan worden de nulpunten bepaald door de factoren waar $x$ in voorkomt.\n\n**Voorbeeld:**\nNeem de vergelijking:\n$$-\\frac{1}{2}(x-5)(x+6)=0$$\nOm de nulpunten te vinden, stel je de vergelijking gelijk aan nul. Omdat het product van factoren alleen nul kan zijn als één of meer van die factoren nul zijn, kijk je alleen naar de delen met $x$:\n* $x-5=0 \\implies x=5$\n* $x+6=0 \\implies x=-6$\n\nDe constante factor $-\\frac{1}{2}$ beïnvloedt de nulpunten niet. Deze factor bepaalt alleen de vorm van de parabool: of het een dalparabool (als de factor positief is) of een bergparabool (als de factor negatief is) is, en hoe 'steil' de parabool is. De ligging van de snijpunten met de x-as blijft hetzelfde.",["D",1773447739061],["D",1773447739383],"35e8e19e-f970-4ab2-aaf2-a37d8011ae0c","wanneer-draait-teken-om-bij-ongelijkheden","Wanneer draait het teken om bij het oplossen van ongelijkheden?","Wanneer draait teken oplossen ongelijkheden","Het teken van een ongelijkheid draait om wanneer je beide kanten van de ongelijkheid vermenigvuldigt of deelt door een **negatief getal**. Dit is een cruciale regel om te onthouden, want als je dit niet doet, klopt je oplossing niet.\n\n**Waarom draait het teken om?**\nDenk aan een eenvoudige ongelijkheid: $2 < 5$. Dit is waar.\nAls je beide kanten vermenigvuldigt met een positief getal, bijvoorbeeld 3:\n$2 \\times 3 < 5 \\times 3 \\Rightarrow 6 < 15$. De ongelijkheid blijft waar en het teken blijft hetzelfde.\n\nMaar als je beide kanten vermenigvuldigt met een negatief getal, bijvoorbeeld -1:\n$2 \\times (-1)$ en $5 \\times (-1)$ worden $-2$ en $-5$.\nNu is $-2$ *groter* dan $-5$. Dus, om de ongelijkheid waar te houden, moet het teken omdraaien: $-2 > -5$.\n\n**Voorbeeld:**\nStel je hebt de ongelijkheid $$-3x < 9$$.\nOm $x$ te isoleren, moet je beide kanten delen door $$-3$$. Omdat $$-3$$ een negatief getal is, moet je het ongelijkheidsteken omklappen.\n\nStap 1: Begin met de ongelijkheid:\n$$-3x < 9$$\n\nStap 2: Deel beide zijden door $$-3$$. Omdat je deelt door een negatief getal, draai je het ongelijkheidsteken om (van $<$ naar $>$):\n$$\\frac{-3x}{-3} > \\frac{9}{-3}$$\n\nStap 3: Vereenvoudig de uitdrukking:\n$$x > -3$$\n\nDe oplossing voor de ongelijkheid $$-3x < 9$$ is dus $$x > -3$$. Dit betekent dat elke waarde van $x$ die groter is dan $-3$ de oorspronkelijke ongelijkheid waar maakt.",["D",1777072735394],["D",1777072735657],"d2501d80-24aa-432a-bfe4-27c59204d555","hoe-interpreteer-je-ongelijkheden-in-een-grafiek","Hoe interpreteer je ongelijkheden in een grafiek?","interpreteer ongelijkheden grafiek","Hoe interpreteer je ongelijkheden in een grafiek, en hoe bepaal je of een lijn 'boven' of 'onder' een andere lijn ligt om de oplossing te vinden?","Om ongelijkheden in een grafiek te interpreteren, kijk je naar de relatieve posities van de lijnen ten opzichte van elkaar. Dit helpt je te bepalen voor welke waarden van de variabele de ongelijkheid geldt.\n\nHier is hoe het werkt:\n\n1. **Begrijp de ongelijkheid:**\n * Als je een ongelijkheid hebt zoals $A > B$, betekent dit dat je zoekt naar de punten waar de grafiek van $A$ **boven** de grafiek van $B$ ligt.\n * Als je een ongelijkheid hebt zoals $A < B$, betekent dit dat je zoekt naar de punten waar de grafiek van $A$ **onder** de grafiek van $B$ ligt.\n\n2. **Vind het snijpunt:**\n Het snijpunt van de twee lijnen is cruciaal. Op dit punt zijn de waarden van beide functies precies gelijk ($A = B$). Dit punt verdeelt de grafiek in gebieden waar de ene functie groter is dan de andere.\n * **Voorbeeld:** Stel Sophia verkoopt ijsjes. Haar opbrengst ($R$) is $2.25q$ (waarbij $q$ het aantal verkochte ijsjes is) en haar kosten ($K$) zijn $1.15q + 352$. Ze maakt winst als haar opbrengst groter is dan haar kosten, dus $R > K$.\n * Om het snijpunt te vinden, los je de vergelijking $R = K$ op:\n $2.25q = 1.15q + 352$\n $1.1q = 352$\n $q = \\frac{352}{1.1}$\n $q = 320$\n * Bij $q = 320$ ijsjes zijn de opbrengst en de kosten gelijk (dit is het 'break-even point').\n\n3. **Interpreteer de grafiek:**\n Nadat je het snijpunt hebt gevonden, kijk je naar de grafiek om te zien waar de lijn van de ene functie boven of onder de lijn van de andere functie ligt.\n * In het voorbeeld van Sophia: we willen weten wanneer $R > K$. Dit betekent dat we zoeken naar het gebied waar de grafiek van de opbrengst ($R$) **boven** de grafiek van de kosten ($K$) ligt.\n * Als je naar de grafiek kijkt, zie je dat de lijn van $R$ boven de lijn van $K$ ligt als $q$ groter is dan het snijpunt ($q = 320$).\n * De conclusie is dan: Sophia moet meer dan 320 ijsjes verkopen om winst te maken ($q > 320$).\n\n**Lijnen tekenen om te interpreteren:**\nSoms moet je zelf de lijnen tekenen om de ongelijkheid te kunnen interpreteren. Om een rechte lijn te tekenen, heb je minimaal twee punten nodig. Je vindt deze punten door willekeurige waarden voor $x$ (of $y$) te kiezen en de bijbehorende $y$ (of $x$) te berekenen.\n\n* **Voorbeeld:** Om de lijn $2x - 5y = 8$ te tekenen:\n * **Kies $x = -1$:**\n $2(-1) - 5y = 8$\n $-2 - 5y = 8$\n $-5y = 10$\n $y = -2$\n Dit geeft het punt $(-1, -2)$.\n * **Kies $y = 0$:**\n $2x - 5(0) = 8$\n $2x = 8$\n $x = 4$\n Dit geeft het punt $(4, 0)$.\n * Met deze twee punten $(-1, -2)$ en $(4, 0)$ kun je de lijn tekenen en vervolgens ongelijkheden interpreteren die met deze lijn te maken hebben.\n\nDoor deze stappen te volgen, kun je ongelijkheden in een grafiek correct interpreteren en de oplossing bepalen.",["D",1775777004754],["D",1775777005029],"3c30e11a-a22a-462c-ba8b-d81e770a2a8a","ongelijkheden-getallenlijn-balansmethode","Hoe los ik ongelijkheden op met een getallenlijn en balansmethode?","los ongelijkheden getallenlijn balansmethode","Het oplossen van ongelijkheden met een getallenlijn en de balansmethode is een handige manier om de oplossing visueel te maken. De balansmethode wordt gebruikt om de 'scheidingspunten' te vinden, dit zijn de x-waarden waar de twee kanten van de ongelijkheid aan elkaar gelijk zijn. De getallenlijn helpt je vervolgens om te bepalen welke intervallen de oplossing vormen.\n\n### De balansmethode bij ongelijkheden\nDe balansmethode gebruik je om de kritieke punten te vinden. Dit zijn de x-waarden waar $f(x)$ en $g(x)$ aan elkaar gelijk zijn, oftewel waar de grafieken elkaar snijden.\n\nDe stappen zijn grotendeels hetzelfde als bij vergelijkingen:\n1. **Neem de ongelijkheid over.** Bijvoorbeeld: $f(x) > g(x)$.\n2. **Maak er eerst een vergelijking van:** $f(x) = g(x)$.\n3. **Los deze vergelijking op met de balansmethode** om de x-waarden te vinden waar $f(x)$ en $g(x)$ gelijk zijn. Dit zijn je 'kritieke' punten.\n\n**Belangrijk verschil bij ongelijkheden:**\nAls je tijdens het oplossen van een ongelijkheid beide kanten vermenigvuldigt of deelt door een **negatief getal**, moet je het ongelijkheidsteken omdraaien! Dit is cruciaal om de juiste oplossing te krijgen.\n\n### Met een getallenlijn\nZodra je de 'kritieke' x-waarden hebt gevonden (met de balansmethode), kun je deze gebruiken op een getallenlijn:\n1. **Teken een getallenlijn** en markeer hierop de gevonden 'kritieke' x-waarden. Deze punten verdelen de getallenlijn in verschillende intervallen.\n2. **Kies testpunten:** Kies in elk interval een willekeurige x-waarde (een 'testpunt') en vul deze in de oorspronkelijke ongelijkheid in.\n3. **Controleer de ongelijkheid:** Kijk of de ongelijkheid waar is voor dat testpunt. Als de ongelijkheid waar is, dan is het hele interval waar dat testpunt in ligt, onderdeel van de oplossing.\n4. **Markeer de oplossing:**\n * Als de ongelijkheid een strikt teken heeft ($<$ of $>$), gebruik je een **open bolletje** bij de kritieke punten op de getallenlijn, omdat deze punten zelf niet meedoen.\n * Als de ongelijkheid een 'gelijk aan' teken heeft ($\\le$ of $\\ge$), gebruik je een **dicht bolletje**, omdat de kritieke punten dan wel meedoen.\n * Kleur vervolgens de intervallen in die voldoen aan de ongelijkheid.",["D",1773447601292],["D",1773447601575],"087e2ad2-1dd3-450f-ba8c-4f8b19861ae4","hoe-bereken-je-de-y-coordinaat-van-een-snijpunt","Hoe bereken je de y-coördinaat van een snijpunt?","bereken ycoördinaat snijpunt","Hoe bereken je de y-coördinaat van een snijpunt als je de x-coördinaat en de vergelijkingen van de lijnen weet?","Om de y-coördinaat van een snijpunt te berekenen, volg je deze stappen:\n\n1. **Vind de x-coördinaat:** Zorg ervoor dat je eerst de x-coördinaat van het snijpunt hebt berekend. Dit doe je meestal door de vergelijkingen van de twee lijnen aan elkaar gelijk te stellen en op te lossen voor $x$.\n2. **Vul de x-coördinaat in:** Kies één van de oorspronkelijke vergelijkingen van de lijnen. Het maakt niet uit welke je kiest, want op het snijpunt hebben beide vergelijkingen dezelfde $y$-waarde voor die specifieke $x$-waarde.\n3. **Bereken de y-waarde:** Vul de gevonden x-coördinaat in de gekozen vergelijking in en reken de bijbehorende y-waarde uit.\n\n**Voorbeeld:**\nStel, je hebt de x-coördinaat van een snijpunt gevonden, bijvoorbeeld $x = 1.5$.\nEén van de oorspronkelijke vergelijkingen van de lijnen is $y = 0.8x$.\nOm de y-coördinaat te vinden, vul je $x = 1.5$ in deze vergelijking in:\n$y = 0.8 \\times 1.5$\n$y = 1.2$\nDe y-coördinaat van het snijpunt is dus $1.2$.",["D",1775777005363],["D",1775777006312],"80e54b46-8112-4fee-9427-2cc5a49d22db","wat-betekent-een-ongelijkheid","Wat betekent een ongelijkheid?","betekent ongelijkheid","Een ongelijkheid is een wiskundige uitspraak die aangeeft wanneer de ene uitdrukking groter of kleiner is dan de andere, in plaats van precies gelijk. Waar je bij een gewone vergelijking (zoals $f(x) = g(x)$) zoekt wanneer twee uitdrukkingen precies gelijk aan elkaar zijn (denk aan de snijpunten van grafieken), kijk je bij een ongelijkheid naar wanneer de ene uitdrukking groter of kleiner is dan de andere.\n\nDe tekens die hiervoor worden gebruikt zijn:\n* $>$ (groter dan)\n* $<$ (kleiner dan)\n* $\\ge$ (groter dan of gelijk aan)\n* $\\le$ (kleiner dan of gelijk aan)\n\nAls je bijvoorbeeld $f(x) > g(x)$ ziet staan, dan zoek je naar alle $x$-waarden waarvoor de grafiek van $f$ boven de grafiek van $g$ ligt. Als er $f(x) < g(x)$ staat, zoek je naar de $x$-waarden waar de grafiek van $f$ onder die van $g$ ligt.\n\nHet gaat er dus om een bereik van waarden te vinden, in plaats van één of enkele specifieke punten.",["D",1765756863743],["D",1765756863994],"0553f851-5ba2-4308-a8b4-a8278aaf4b53","hoe-bepaal-je-wanneer-lineaire-groei-groter-is-dan-ander","Hoe bepaal je in welk jaar lineaire groei groter is dan een andere?","bepaal welk jaar lineaire groei groter","Hoe reken ik bij lineaire groei uit in welk jaar het een groter is dan het ander?","Om bij lineaire groei uit te rekenen in welk jaar het ene groter is dan het andere, maak je gebruik van lineaire ongelijkheden. Dit betekent dat je twee lineaire functies met elkaar vergelijkt om te bepalen wanneer de waarde van de ene functie die van de andere overtreft.\n\nJe doorloopt hiervoor de volgende stappen:\n\n1. **Stel de ongelijkheid op:** Begin met het opstellen van een ongelijkheid die de situatie beschrijft. Dit houdt in dat je de twee lineaire groeiformules (bijvoorbeeld $y = ax + b$) tegenover elkaar zet met een ongelijkheidsteken ertussen. De veelgebruikte ongelijkheidstekens zijn:\n * $<$ (kleiner dan)\n * $>$ (groter dan)\n * $\\le$ (kleiner dan of gelijk aan)\n * $\\ge$ (groter dan of gelijk aan)\n\n *Voorbeeld:* Stel, je hebt twee situaties:\n * Situatie A: $y_A = 10x + 50$ (waarbij $x$ het aantal jaren is en $y_A$ de waarde van situatie A)\n * Situatie B: $y_B = 15x + 20$ (waarbij $x$ het aantal jaren is en $y_B$ de waarde van situatie B)\n Als je wilt weten wanneer situatie A groter is dan situatie B, stel je de ongelijkheid op: $10x + 50 > 15x + 20$.\n\n2. **Los de bijbehorende vergelijking op:** Om het 'omschakelpunt' te vinden, los je eerst de vergelijking op die hoort bij de ongelijkheid. Je vervangt het ongelijkheidsteken dus tijdelijk door een is-gelijk-teken (=). Dit snijpunt is het moment waarop de twee precies aan elkaar gelijk zijn.\n\n *Voorbeeld:* Voor de ongelijkheid $10x + 50 > 15x + 20$ los je eerst de vergelijking op:\n $10x + 50 = 15x + 20$\n $50 - 20 = 15x - 10x$\n $30 = 5x$\n $x = 6$\n Dit betekent dat na 6 jaar de waarden van situatie A en situatie B precies gelijk zijn.\n\n3. **Bepaal de oplossing van de ongelijkheid:** Nadat je het snijpunt hebt gevonden, kijk je wanneer de ene functie daadwerkelijk groter is dan de andere. Dit kun je doen door een waarde te testen die voor of na het snijpunt ligt, of door naar de grafieken van de functies te kijken.\n\n *Voorbeeld:* We weten dat bij $x=6$ de waarden gelijk zijn. We willen weten wanneer $10x + 50 > 15x + 20$.\n * Test een waarde kleiner dan 6, bijvoorbeeld $x=5$:\n * $y_A = 10(5) + 50 = 50 + 50 = 100$\n * $y_B = 15(5) + 20 = 75 + 20 = 95$\n Hier is $100 > 95$, dus $y_A > y_B$ als $x=5$.\n * Test een waarde groter dan 6, bijvoorbeeld $x=7$:\n * $y_A = 10(7) + 50 = 70 + 50 = 120$\n * $y_B = 15(7) + 20 = 105 + 20 = 125$\n Hier is $120 < 125$, dus $y_A < y_B$ als $x=7$.\n\n Uit de testwaarden blijkt dat situatie A groter is dan situatie B wanneer $x < 6$. Omdat $x$ vaak staat voor jaren en niet negatief kan zijn in dit soort contexten, is de oplossing $0 \\le x < 6$. Als de vraag specifiek is naar 'in welk jaar', dan zijn dit de jaren 0, 1, 2, 3, 4 en 5.\n\nDoor deze stappen te volgen, kun je bepalen in welke periode of welk jaar de ene lineaire groei groter is dan de andere.",["D",1779405373672],["D",1779405373904],"sectionUuid","62ff2345-7730-43a2-947f-e29ecf2f06e2","sectionTitle","5.3 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